What the...

My first thoughts. Who is gonna use a 5 star cautery sword. Okay, maybe for Ice queen? Hell no, combuster works well in Ice queen. Better yet, wear a full skolver set, added with BTS and you are gonna get a full damage boost on swords. The cautery would be completely useless.

So you want another Cold Iron Vanquisher with a Slime bonus? Nah. I don't see why they added CIV in first place if there's so much elemental weapons. The same counts for this and shadow.

I agree that Cautery Sword needs some love, but making it Pierce or some such isn't going to make it on par with the rest of the Brandish line. I think what it needs is to be taken out of the line entirely and given its own line.
The things that makes the Cautery sword unique is that it does massive damage to a monster family but without an element attached, making it still useful in every other situation. Work off of that and expand on it.

Here's my take on it:
Have Cautery be the start at 3* and have three branches, each dealing increased damage to two monster families each to cover them all. They would all still have Normal damage and still be useful as against the other monster families, making them quite useful for levels with mixes of monsters in them and for players who don't have full weapon slots.
So if you run into a Fiend and a Jelly on the same level, a Pierce or Shadow would work on one, but not the other. Unlike a Cautery sword which could still kick both of them up and down the Clockworks.

Anyways, that's my idea for it. I agree with you that the sword does need some love... actually a LOT of equipment does. So +some number and what not, I support.

Maybe use opposing families then? Like that Fiend and Jelly example I put up. They're resistant to the type that the other is weak against, so would be kind of useful. If we go with that it would be:
Fiend - Jelly
Beast - Construct
Gremlin - Undead

And if the usefulness of such a line of swords doesn't compare to that of the Brandish, maybe even go so far as to make it work on three families. Like:
Slime - Fiend - Undead
Gremlin - Construct - Beast
Or some other combination of the three.

The biggest problem I foresee with such a line of swords would begin to encroach upon the Leviathan/Cold Iron Vanquisher of being a all around great weapon. These would be more specific, but they would still have to be weaker in damage dealing in comparison to the Levithan/Cold Iron unless it's attacking the appropriate family.
Nevertheless, it's something to look into. I might draw up some designs myself if inspiration hits me.

Not necessarily. Just removed from the Brandish line and given it's own. Having it start at 3* avoids the problem of having to make a new 2* version and having to give people with a Cautery sword already something to make up for deleting it. It would still exist in all its awkward slime hating glory, but on its own without its annoying showy parent Brandish.

Underpants fairies.

If we want to keep old Cautery in the game it would either have to function uniquely from its siblings, be an unoriginal repeat, or become a completely separate alchemy line which would open a lot of opportunities.

One of the appeals to Vanquisher is its charge attack knockback being lower than Leviathan. It helps when you want to use it in parties without throwing enemies at your allies further away.

Someone give this guy a medal.