Yeah, personality tags, chosen during character creation (e.g. as part of the chosen background as per 5e) and/or gained during play, would be very nice. Solasta is a good example of how it can be done.
Certainly chosen at character creation. Like, again, in Solasta, where you have a good mix from your alignment and background (and something else could be thrown in, race/class? Just choose from all existing?). Acquired during gameplay sound great as well, but it's trickier; the game doesn't know your character's true motivations, only you do. So your PC could get an "altruistic" tag for helping an orphan with "and you'll repay me later, with interest" mindset. But if a good way could be found to implement this, I'm all for including acquired personality tags.
Personality tags are great if the developers actually use them. If they pop up only once or twice during a playthrough don't bother. If they intend to do it they should do it right
By all means. The problem with this in BG3 is that it's a cinematic AAA game with tons of stuff in it. And if they were to do personality tags justice, it would be a lot of work/resources. Unfortunately, this is the price a game pays for full voice acting and lots of dialogue lines. But I think it would be worth it in the end. Getting custom characters right and making them feel alive is very important in a D&D game.