Hi! Currently, I say that all Normal damage weapons are underused. Sure, specialized damage can get you max DPS, but so can normal weapons. Wait, what?
For easy reference, Special damage means Piercing, Elemental, or Shadow. A weapon type means the types sword, gun, and bomb.
When to USE
Normal damage is good when you have limited weapon space. That goes without saying. If I only have two weapons, perchance I'm not going to be equipping two weapons of the same damage type, even if they are of different weapon types.
That brings me to my second point: even if you have a Special damage type, unless you are equipping two weapons of the same type, you might want a Normal damage weapon.
Lastly, it may be my personal preference to use a particular weapon type against all enemies in a single level, even if I only have one slot for that weapon.
For example, (This is a real example) I have just two weapons, a sword and a gun. I'm entering a Wolver Den. I want to use a gun for Beasts, so I take a Callahan, one of THE best guns for beasts, IMO. Then, in case I see any Polyps and Lumbers, I should take a Gran Faust, right?
WRONG! If I see a Silkwing, my sword won't do squat unless I beat the snap out of the Silkwing, but that takes too long! My gun doesn't do enough against Silkwings in a short enough time span. I want to be beating down the Wolvers instead of Silkwings. Well, why don't I take a piercing sword? But then what about polyps and lumbers? Well, I could take Umbra Driver or Toxic Catalyzer, but what if I really want to have the Callahan against beasts? Therefor, that leaves me back at the sword. I am stuck with a Normal damage sword, which is fine.
The fact is, taking a Normal sword is optimal because I want to be able to use it against all that can be thrown at me.
When NOT to use
If you have more than one of the same weapon type, having either one deal normal damage is inefficient unless it is your top preference to have that weapon in that damage type. For example, you love the Pulsars. So, you take two. (Supernova and Polaris) You can have all the Pulsar fun you want.
If you are fighting enemies which share weaknesses, then it's obvious to take just one damage type.
If you have many weapon slots or you don't care about what you use on what, and you are happy as long as you are getting criticals, you are likely just to take what is more effective for that level.
If you are in a party with players who's weapons make up for the lack of, let's say, having short range, then having just swords is very very viable. When in parties, the weaknesses of having just one weapon type are minimized. (As long as the people making up for your weaknesses are alive!)
Familly bonuses: why they are underused
When using just one weapon type, family bonuses are useless because they only effect one monster family. If you used a bonus just for that weapon, you could use whatever grants that bonus on any enemy and still get equal boost.
However, there is a flaw with doing the above. If you take just one weapon type, you won't have what is most effective for every situation. If you add another weapon type, it will suddenly lose out because it doesn't get the bonus.
This is where monster family bonuses come in. You can deal increased damage with any weapon, and often you only need one or two items to do this. (And you don't need Chaos.) Every level is themed for a particular enemy and that enemy is usually the predominant type, so you can deal with these enemies very effectively. If you can use all weapons properly, you are free to pull out any weapon you need for a particular case. My favorite is to use a gun during the approach, hopefully to keep the enemy flinch-locked or dodging, and then they are prime targets for up close assaults. (Which was made more viable because of the dash'n'bash moves.)
One thing which skews this whole thing is bombs, because CTR makes them many many times more effective. You'll likely want bomb CTR if you use bombs often at all. Fortunately, a single Volcanic Demo or Elite Boom Focus Mod can make up for this. (Though earlier on you are more limited.)
Conclusion
It seems that, despite popular recommendations, specializing in one weapon type isn't the most effective strategy. Your DPS could be far higher if you are more open minded. Yes, sometimes swords are most effective, but other times bombs and guns are much better, even if not safer. Bombs get the crowd faster as a whole, and guns can deal damage and range efficiently, or can pick off and prep enemies and mobs for a brutal assault.
Chose your weapons wisely, knights, and go forth into the depths! Do not be linear, because you could have much more fun otherwise.
Yes, this is all very reasonable. In fact, I say pretty much the same thing in the "Mixing Weapon Types" section of my sword guide (without going into such a detailed example).