I would like to open this by talking about Family Damage Bonuses:

Family Damage Bonus should stack on top of Overall Sword/Handgun/Bomb Damage Bonus to a total of +9
Family Damage Bonuses cap at +9 while overall Damage Bonuses cap at +6
Family Damage Bonus can stack on top of Overall Damage Bonuses to go higher than +6
Low (+1)
Medium (+2)
High (+3)
Very High (+4)
Ultra (+5)
Overdrive! (+6)
Colossal! (+7)
Monumental! (+8)
Maximum! (+9)

Why?

This would change the whole idea of Family Damage Bonuses being an unnecessary or mostly useless UV since the best armors for damage already offer a close to, if not, +6 Overall Dmg Bonus (with Pet perks or trinkets added as well), thus rendering Family Damage Bonuses useless. Since weapons cap at 3 UVs having 6 Variants that would actually not be so useless would also make players consider which UVs would be best for their weapons. This would also make normal damage weapons with incorporated Bonuses better since they would bump the +6 damage bonus to +9 (Cold Iron Vanquisher with the High Undead; Wild Hunting Blade with the High Beast; you get the idea).

Of course, this would also work backwards, with Family Damage Penalties stacking on top of Overall Damage Penalties to a total of -9; if they ever add items in the future which include Overall/Sword/Handgun/Bomb Damage Penalties, that is.

Ok, but wouldn't it be too op, you would have an immensely powerful hit against certain enemies?

And to that I say no. The meta as it is now is basically using CTR and ASI to shred enemies with quick succession attacks and quickly charging the weapon to do it over and over. This would make some people rely less on CTR or ASI to instead focus on more powerful attacks using Family Bonuses. Sure you could get a CTR VH; ASI VH; vs Family VH but 1st: it would take a lot of effort just to get 1 weapon to be like that and 2nd: you could instead opt for another bonus against another Family instead of CTR/ASI and thus make it more effective on one area only. In case you ditch CTR then your ASI would complement your 2 other family UVs (which would presumably be for the 2 families that are weak to the weapon's dmg type) and your quick strikes would be effective and strong. Sure you could go for a charge but it would be way slower and not very efficient to use. The same would happen vice-versa. If you instead had CTR and the 2 family bonuses, then your quick Charge Rate would make it very efficient to deal a good ton of damage on enemies since your normal attacks would be considerably slower.

Well then, but how about a Black Kat Set; Swiftstrike Buckler; Swift Steps II; and ASI +3/+4 UV & CTR +4 UV (+ Family Bonus High) on all weapons? Wouldn't that just make it even stronger?

And sure, it would make them stronger but they would still have the option between 2 Family Bonuses + CTR/ASI or keeping ASI & CTR + 1 Family and it would give other armors in the game a chance to compete against these meta sets.
There's also the fact that with the Shadow Lair buff to T4, even with those sets, people struggle doing them (which is intended, I understand) and these changes wouldn't make it suddenly easy but weapons would be a little bit more effective against these crazy strong enemies.

In case you're still not convinced, let me present you my next point:
More UVs and more Abilities for armor and weapons.

Abilities are an important point. Most of what people use are the Chaos Set / BK Set abilities since those provide strong and effective bonuses to kill enemies fast before they overwhelm players. Especially BK, with its MSI +2 and Dmg Bonus +6 (total) to EVERY WEAPON against ANY MONSTER. New Armor sets with the following Abilities would surely make BK more debatable to use in certain situations, depending on the combination, of course. Some examples of these new abilities would be stuff like:

Charge Time Increase (Overall/Bomb/Sword/Handgun);
Damage Penalty (Overall/Sword/Handgun/Bomb/vs Family (except Fiend cuz thats already in the game; Yes, I'm looking at you Fallen Set));
Dash Cooldown Reduction;
Dash Cooldown Increase;
Shield Regen Increase;
Shield Regen Decrease;
Shield Respawn Time Increase;
Shield Respawn Time Decrease;
Bash Damage Bonus;
Bash Damage Penalty;
Bash Consumption Reduction;
Bash Consumption Increase;

(All of these cap at +6/-6, as usual)

You might be thinking "wow, these are a lot of new abilities but how could they be on pair with BK?"

Consider the following set:

Sample Helmet/Armor:

(Ignore Defense Power)

Status Resist
(Nothing)

Abilities
Sword ASI: High
Sword CTR: High
Sword Dmg Bonus: Med
MSI: Med
Bomb Damage Penalty: High
Handgun Charge Time Increase: Low
Handgun Attack Speed Decrease: Low

Would you look at that! Just slap on a Sword Dmg Bonus +2 trinket and you're maxed for swords. Put Swift Steps 2 and you're at MSI +6. Assign some Family UVs to weapons and you got +7/+8/+9 on them and dont forget that you still have 1 trinket slot to put what you want and No Status Effect Penalties!

Now before you come after me saying that it was a bad example as it was too op or badly balanced just take a moment to think that it was just to show you how you could tinker around with those negative and positive abilities and have sets on pair with BK while also having reasonable drawbacks to make up for the huge boost. My point was that by introducing these new Negative abilities, you would have a lot more room to make distinct and yet powerful sets worth using instead of BK. BK might be good against vanaduke and a whole lot of other places but Sample Set might be better at other stuff where BKs is weak at. And this is only in the Armor department; Weapons and Shields could have imbedded Penalties or Bonuses or both and more!

"Alright, fine. But where could BK be weak at?" you ask.

This brings me to my next topic... But before that, since we're on the topic of UVs and Abilities, let me just show you some new stuff that could be interesting to add:

New Shield UVs:

Shield Regen Increase: Low to Very High;
Shield Respawn Time Increase: Low to Very High;
Bash Damage Bonus: Low to Very High;
Bash Consumption Reduction: Low to Very High.

New Armor UVs:

Dash Cooldown Reduction: Low to Very High;

Oh and before we move onto the next topic i just wanted to end this one with this:

Imo, Overall ASI should work on bombs and they should get an ASI UV added as well.

When? You only have to charge the weapon to make them work... And that is right, except that "attacks" on bombs occur after the charge: Here's my thought process:

When you hold the button, you are charging, therefore CTR for any weapon including bombs is when you hold the button. Pretty straightforward and obvious. But ASI is a little different. If you look at the tooltip of a bomb, you see the "attack speed" which is just the time that it takes for the bomb to explode. If you look at that bar on a sword/handgun, the same bar, corresponds to the time for a normal attack to take place and ASI affects that, therefore, ASI should affect Bomb's attack speed as well. Just a thought. I thought of maybe giving it another name like, I don't know, Explosion Speed Increase but that would separate bomb "ASI" from handgun/sword ASI, thus Overall ASI wouldn't work on bombs.

Good. Now that, that's out of the way, let's resume the BK topic...

Where could BK be weak at?

Tinkering around with abilities and new UVs should be enough to produce something that would question the superiority of BK over the other sets.

But because i have more stuff to show just in case it serves as an idea for other people, let me introduce my ideas of content that I personally would find very interesting to add into the game as to complement the ideas mentioned above:

First Idea: Silver Enemies

These are based on Gold Enemies (the type of monster that is invincible) but instead of being invincible, they work differently.
When they spawn, they are damageable and you need to kill them just like any other monster. Once they reach 0 hp is when the magic happens.
Instead of dying they turn invincible and keep on fighting alongside the other monsters in the battle. They only truly die and drop their stuff when all the mobs of the wave are killed, including Silver enemies.

This type of monster would add an interesting difference to regular battles. As you all know, by killing the stronger enemies you are left with the weaker enemies and the battle gets easier and easier the more you kill. With these monsters, that wouldn't be the case. You would have to kill all of them first and only after would they all die. This would make the battle maintain its difficulty throughout its course until the end, making a strong Offensive probably not the most viable solution and instead make people opt more for Defensive sets worth using as to not die quickly.

Silver enemies would be resistant to all types of damage except normal and would be "Hardened" by default (you will get what I mean if you keep reading), have a higher defense and an unusual resistance to all status effects.

Second Idea: Levels with a lot of enemy diversity. I'm talking Status effects and Families.

These would be levels in which you would have all types of monster families and monsters with all types of status effects. Ice, fire, poison, freeze, curse, etc all in the same level. Traps of all kinds as well. At first I know that this just sounds as a stupid idea. Just a bunch of enemies put together in a level at random. But the randomness is what would be special about them.

One of the major things would be that Normal Damage weapons would shine in these levels. Not knowing what to come, what to expect, are you gonna take something that might not work well on enemies or worse, boost their powers?
Another major aspect would be the fact that sets like Chaos or BK would be terrible in terms of defensive power. So many status penalties that if you were to be affected by status effects, their length really wouldnt make it easy on you. And don't forget, taking 3* shields to boost your stats wouldn't be very wise either as enemies could easily break it and affect you with tons of status effects, especially with so many different types of monsters.

Before I continue onto the next idea, let me just have a word on Normal Damage Weapons:

As you all know, they deal normal damage to every mob. Never critical nor weak. As a weapon that only deals medium damage, never weak, with their drawback being that they also never deal Critical damage they ought to be somewhat good, right? Why are they so rarely used by players? And the answer to that question, I would say, is the lack of variety of families in a given level.

There are many levels, each with their unique enemies. Some have a bit of every enemy family, most have almost exclusively one family in the whole level and while I see that as a good thing, the number of levels that display around 2 more predominant families for the duration of the whole level far outweigh the ones without. Considering there are only 2 families in a given level, if their weaknesses are different, all you gotta take with you are 2 different weapons and you're good to go for the level, making Normal Damage weapons not really worth taking as they will mostly do worse than better. Add the ability to take 4 weapons and you won't ever need to touch a normal damage weapon again.

And that's not the only thing. Normal damage weapons, for the most part don't even have good medium damage or charges worth using. In the case of Amputator, it does have a good charge but if you look at the Damage that it does, it's a lot less than its Elemental/Shadow Counterparts' Medium damage when it should be about the same or maybe a little bit more as it can't deal Criticals, because, again, Normal Damage weapon's imbedded drawback.

Other than Amputator, Normal Damage Swords' charges, especially, have very weak and/or do not hit a lot of targets.

Leviathan's or CIV's charge is a circle around its user, which can prove itself useful in some situations but its not that likely that you will be surrounded by enemies the moment you decide to do a charge attack. This is just one of many examples.

So what can be done about these weapons?

First of all, the idea that should've been implicit from what I said previously:

-Fixing the damage output of Normal damage weapons;
-Fixing the charge of normal weapons for attacks a bit more powerful and that can hit a lot of enemies at the same time, or atleast if their charge is small like CIV's, bump up their charge damage by a LOT to make it worth the strike;

Second, in addition to the idea above, this idea splits into 2 sub-ideas:

1: Adding a Critical chance for normal damage weapons to deal Critical Damage, just like a damage-specific weapon does on a weak enemy. Not too much to not make it overpowered but also not too little to not make it not worth existing in the first place.

This could also introduce a new UV for Normal Damage weapons to increase Critical Chance by a bit.

OR

2: Adding a special passive or active perk/ability to normal damage weapons as to distinguish them from specific damage weapons and no, I don't mean the family damage boosts.
Some examples would be: MSI +3 when holding (Wild) Hunting Blade or perhaps it having a chance to inflict you with a speed orb for X seconds every hit (hitting fast and repositioning yourself feels so good that i wish the blade had this);
Cold Iron Vanquisher Line chaining enemies that are hit by the charge attack, imobilizing them and lowering their attack damage or something of the sort.
The Spur line creating more disks if you pair it with a swiftstrike buckler (just because they look like they belong to the same set);
I don't know, something creative and useful.

This next idea is not much about the Normal Weapons themselves but still compliments them in an indirect way.
In addition to all the ideas above:

-Adding new monsters that would be strong to everything except normal damage. Kinda like Tortodrones but instead of being mini bosses, they would just be normal enemies.
-Adding new monsters that would be strong to 2 damage types and weak to 1; meaning 2 damage types, lets say: shadow and elemental, would deal weak damage while piercing dealt strong damage. (Possibly even new enemy families for every combination.) Likewise, there could also be new mobs/families which would be weak to 2 damage types and strong to 1, just to balance things out.

Third Idea: Hardened Enemies (I don't have a better name for them)

These would be enemies that would be exactly the same as their normal counterparts but wouldn't be able to take in too much damage in a short period of time or they would shield all additional damage (you know, do the purple glass sound hit thing). The zombies in FSC are a good example. People use 3 or 4 brandish charges on top of them at the same time and they dont even have time to react before being sent to oblivion. This would make them shield maybe 3 of the 4 charges once too much damage got registered at the same time. Note that Status effects would still take place even when the damage was shielded. Autoguns wouldn't be able to kill a Trojan in one go either. Maybe the first 7-9 shots (of a charge) would be registered but the ones after would just be ignored if the HP of the trojan had dropped more than 67%-75% in less than a certain time frame.

Hardened enemies would have around a 1 in 4 chance to replace a normal enemy and 1 in 2 to replace a giant (trojans, guards, etc). Bosses & their minions would never have this effect, nor would Void enemies, SL enemies or lesser enemies such as Silkwings and the like.

And lastly, to sew everything together would be for the aforementioned low family diversity levels, to have a decent number of All Resist/2 Resist 1 Weak mobs as well as occasional Silver enemies (in arenas), randomly spawning in battles or alongside other enemies just to make Normal Weapons a more viable choice.

The following stuff are just some random gear I put together as examples of what could be added:

Pacifist Hood/Mantle (5*):

Defense Power

Normal - 6 blocks and a bit (.2?)
Elemental - Same as Normal
Shadow - Same as Normal
(6 blocks and a bit refers to the same protection as a Black Cat Mail at lvl 10)

Status Resist

Fire +6
Shock +6
Freeze +6
Curse +7
Poison +6

Abilities

Damage Penalty: High

Health Pips

Level 1+: Health Bonus: +4
Level 5+: Health Bonus: +5

Helmet/Armor of Love or something related to HP (5*):

Defense Power

Normal - Same protection as Almirian Crusader's Normal Prot
Elemental - Same as Blazebreak Armor's Normal prot
Shadow - Same as above
Piercing - Same as above

Status Resist

Curse +4
Poison -4

Abilities

Hearts and Health Capsules all provide +1 Pipe
Charge Time Increase (All): Low

Health Pips

Level 1+: Health Bonus: +9
Level 5+: Health Bonus: +10

That's about it. Thank you for reading!

Edit: Removed the arrows because they made words disappear.