I think that amongst many, Dex builds are regarded as better then Strength builds unless you want to use a two-handed weapon.
Ironically, the rapier actually needs a stronger person to utilise effectively than a Longsword.
Balance wise, I get why it isn't light, but it annoys me nontheless, because the natural accompaniment to a rapier should be a dagger, but you can't dual wield with a rapier without the feat. but again, balance. I get it. And older rapiers were long enough that I can see it (although even then they were often used with a main gouche in the off hand).
Rapiers were paired with the main-gauche, but it was rare that they would be used in simultaneous attacks as per D&D's dual-wielding mechanic. The main-gauche was for defence and for when your opponent was too close for you to employ your rapier effectively. Generally you would be trying to keep at rapier-fighting distance, not dagger-range.
Anyway, if you're cool with full plate in your fantasy, in terms of timeline of equipment in IRL medieval Europe, you should be fine with rapiers. Same amount of anachronism.
I think part of the issue is not the anachronism of them both being around at the same time, its the practicality of trying to use a rapier of said full plate.
Of course all this is rather muddied by the issue that there wasn't really a point at which rapiers just appeared. The longsword was always both a cutting and thrusting weapon, and there was a gradual evolution to accentuate the thrust, with cut-and-thrust swords for civilian use growing longer and thinner to improve their effectiveness against an unarmoured opponent using a similar weapon. The point at which these swords became the D&D rapier isn't definite.
Sort of. The rapier was a civilian weapon. For military use people would still be using what D&D would term a longsword as the backup to their polearm. Rapiers didn't see military use until quite a bit later, after armour was almost fully phased out.
You certainly could do that. I myself wouldn't bother with the change in damage die because I haven't found having a d8 weapon for DEX characters to be an issue in my games. Almost all of my DEX-focused PCs have either gone ranged or dual-wield (which precludes using it anyway until the feat is taken) so having a d8 finesse option I'm okay with. Mainly for DEX-based Edritch Knights and the like, who can't dual-wield if they want to cast spells (without constantly having to sheathe their off-hand weapon.)
For me... it's mainly magic weapon stocking where not needing to split between longswords and rapiers would make me happy.