I've always seen the rapier/broadsword comparison as comparing apples and oranges. The adoption of the rapier wasn't an island - it also had to do with the general abandoning of armour due to the development of firearms, and has to be seen in that context.
In essence, you've got a couple of 'packages' happening historically. First you have the relatively unarmoured rapier-wielder with a pistol on his belt and perhaps a main-gauche or dagger in his off hand. This is a great package for duelling one-on-one or in small groups. But it's a bit of a liability in the crush of close combat, in a mass battle, or in a siege situation, when you're likely to have all sorts of random attacks coming at you from random directions, and you won't be able to parry (or indeed see) them all. In that case, armour will save your butt. And again, in a close chaotic crush, you won't often have time to line up your rapier with a weak point in your enemy's armour, and simply bashing away with a warhammer or similar may well be a better solution. To say nothing about how easily a rapier could get broken in a protracted battle, especially if there's shields and armour involved. Here's where the heavy-armour-and-big-choppy-thing package comes into its own.
It's worth noting that as dangerous as the rapier is in a duelling situation, it was never adopted in a military context, even though melee weapons such as pikes, halberds, cavalry sabres and even zweihanders (in certain cases) remained in military use for a long time after the advent of black powder. Weapons are situational.
And of course that's before bringing a fantasy aspect in to it. However lethal a rapier is to an unprotected human, it's hard to see it bothering an ooze much. Or a zombie. And while a couple of inches of penetration in the right spot will kill a human, that sort of thing'd be much harder to pull off whe you're fighting a frost giant, or a purple worm...