Right click on the name in the chat box and you can 'target' the player if they are close enough.
Mouths move as you speak in game so it's not like they're *just* standing there.
Chat bubbles above the characters would litter the screen with unnecessary clutter. I know, you already said something about that, but still bringing it up.
As for the mod thing, they really don't care unless it gives you an advantage/cheat I believe. At least they don't really enforce action against folks that use mods like that from what I'm aware of.
Tbh also, I'd rather they use time and resources to fix or add other things than this.

Also, if folks be talking like, that much and constantly around you... wouldn't the chat bubbles like really get in the way/possibly overlap? You'd still not be able to read everything that fast if you can't read the chat box that fast, because the chat bubbles would have to disappear for whatever they say next.

This just doesn't seem like it'd be that much of a fix as you think it would, but perhaps I could be wrong.