I rather like rapiers. Probably because I prefer more overland/town adventures, so the "stab Orcus with a fencing weapon" isn't really a problem. Also, my players tend to go with a concept and the mechanics are secondary, so a functional rapier just means that their swashbuckler doesn't suck down the rest of the party.
Now, do I love the 5E handling of the rapier? No. But I don't hate it, either. It's a dex-based long sword. The end. I don't actually see it as a fencing weapon (in the modern sense). It's still a big-boy weapon, whether that means an elven blade or just a broadsword. I'd actually use the long sword stats for a katana.
I don't object to a one handed d8 finesse weapon that isn't light, in fact I would like there to be one that is slashing and one that is piercing (can't see one that is bludgeoning?). I just really dislike the name rapier as it feels anachronistic as compared to the other weapons and my concept of the game.
I'm thinking of renaming the current scimitar as a "cutlass" and making the scimitar a one handed d8 weapon that isn't light and that does slashing damage.
I'm less certain what to call "the piercing weapon" - rapier makes some sense but I dislike it. Maybe longsword and call the current longsword a broadsword? Feels a bit more AD&D ish.
The rapier is only a symptom. The issues driving it's popularity are the dominance of Dex, the need for a finesse weapon to do Sneak Attack damage and the fact that it's the sole finesse weapon that does d8 damage. Hell, I've got a Strength-based barbarian/swashbuckler that uses a Rapier because A) he's a swashbuckler! and B) he can do Sneak Attack damage with it. Don't blame the playa, blame the game.
Yeah, the idea of having a dueling weapon against heavy armored foes, mindless beast, creature as large as a barn is laughable. I did a lot of fencing once and I can tell you that the main reasoning behind the finess weapon being able to to as much damage as a heavier weapon against an armored adversary; that your PC hits between the plates, is strange. There's no way a fighter with a rapier is able to pinpoint 4 times in 6 seconds the small breach in the armor while the foe is defending himself.
Anyway, in my head canon Scimitars are Sabers and Rapiers are Broadswords, longswords are Arming sword, two-handed swords are Longsword.