I play every single table asynchronously (aka turn-based), it works well.
If there is ever any issue about figuring out what your opponent players you can look in the game log, and if that is not enough info (for some games it really isn't), you can hit the log entry right after you finished your previous turn and hit "replay from this point", BGA will re-display all that your opponent did and bring you back to present time where it it your turn. I typically do this for games like Oriflamme and Rainbow, or Fluxx, or Skull.
The most tedious ones are card games because there are a lot of "turns", each card played by each player is a turn, but even so I do play games like Hearts and Solo Whist in turn-based Arena; we can typically get through a full game over the length of an Arena season. Obviously two-player card games are by definition faster and more convenient, like Haggis or Cribbage. I've found 6 nimmt typically goes by at a brisk pace, since every player selects their card at once. On the other hand, Tock games are usually way too long when played async. However if you have a mind for counting cards you might find that much easier in synchronous play, and that's really an important part of Hearts. But you can always use the game log to reach the same conclusions.
Great asynchronous games are games with no interaction, where each player kind does their own thing and compete for final score. Stuff like Can't Stop Express, Welcome To, Railroad Ink, dice games like Martian Dice, Yathzee, Piraten Kapern, Dungeon Roll, Cardiceo.
Other low-interaction games with very manageable turn counts turns include Copenhagen, Patchwork, Conspiracy, KIngdom Builder, Kingdomino, Color Pop, Chromino, Cubirds, Azul.
It can work too for the simple absract games like Connect Four, Pyrga, Squadro, United Square, Bug, etc. but these games might benefit from developping a longer strategy that can be difficult to "fall back into" mentally at a glance, which can be impaired by async play -- but you can always use the "note" tab.
Some coop games can also be fun async, like Tranquility or Forbidden Island; for the latter you can leave messages in chat for whenever your partner takes their next turn. The latter has "out of turn" actions but you can always send a chat to the other player.
One of the very few games that does not work async but yet somehow has it enabled is Solo -- a core part of the game is "jumping in" during someone else's turn if you have the card for it, which is simply not manageable in async. Turn-based play should be disabled, like it is for Coup.
