If the robes aren't gonna get restricted to monks and spellcasters in the future, it'd be pretty amusing (and a little disheartening, perhaps) to see min-maxy Dexterity + Constitution barbarians wearing expensive silks all caked in blood and mud along with a shield and a scimitar or something.
Is that a thing? Barbarians fencing in robes?
Using finesse weapons under rage seems like a Monty Python scene.
It can be. Sort of. Finesse weapons let you attack with Strength
or Dexterity, so there's nothing stopping you from picking up a rapier or a dagger and using it with Strength. It's usually a little pointless, since most finesse weapons do worse or at best equal damage as their non-finesse equivalents.
One odd quirk comes if you're mixing Barbarian and Rogue. One of Sneak Attack's triggers is getting advantage while attacking with a finesse weapon, but it doesn't care about the stat used. Barbarian can give themselves at-will advantage when attacking with Strength, albeit at the cost of giving advantage to attacks against the Barbarian (Reckless Attack feature). In the tabletop, it's a good foundation for a tankier Strength Rogue or a more tricksy Barbarian.
"What if I built around Dexterity?"
It's possible, but there are trade-offs. A Barbarian:
1) does extra damage with Strength attacks while raging
2) doesn't get the advantage from Reckless Attack on non-Strength attacks
3) gets features based around buffing their crits, which synergises better with greataxes and the like
4) needs 13 Strength to multiclass out of Barbarian
So that puts the Dex Barb squarely in the "odd thing to do" category. Still, you never know. Maybe they'll drop in some Belts of Giant Strength somewhere, and my unstoppable naked halfling tank will get their time in the sun.