DEX has always been a god-stat. The problem isn't the rapier, or even dealing damage or hitting things with Dex at all. The problem is that Dex factors into too many stats compared to any other score.
Want to go first? Dex.
Want to avoid getting hit? Dex.
Want to get out of the way of a fireball? Dex.
Want to do any physical task that doesn't rely on brute strangth? Dex.
Want to steal something? Dex.
Want to hide something? Dex.
Want to hit with a bow? Dex.
Want to play a class that isn't a caster or a brute? Dex.
I'm sorry I just don't see how 5E allowing you to deal damage with Dex has made Dex significantly more powerful. And I don't see how it's removal would make it less powerful. Even back in 3.5 when things worked as people above have suggested, Dex was still an incredibly powerful stat, determining a disproportionate number of interactions with the game, often heavily valued interactions (such as reflex saves, AC, and hide/move silently; hey we all know that paladin!).
4E attempted to solve this by allowing two or more stats to provide similar effects. Int was available for both AC and Initiative, for example and Fort AC could key off Str or Con and the ever objectionable damage being dealt by Charisma for paladins! It certainly moved the game towards making more classes SAD, but it dramatically reduced the necessity for people to pump dex.
If you want to fix Dex, start at the root.
All that aside, this has never been a problem at my tables, ever. If you're seeing it as a problem at your tables, I'd love to hear some examples, but generally speaking these sorts of "fixes" come across in response to hypotheticals and white room math.