Well they can be watching all the action and thinking the whole time as the action leads up them, just like we do in poker. To me it's different, subtly perhaps.

For my implementation, I'm passing in the current hand's action history (e.g. Player 1 raises to $X preflop, Player 2 calls, Player 3 calls. Flop is A B C, Player 2 checks, etc) whenever the action is on the player.

Your idea of having it being passed in real time and having the LLM create a chain of thoughts even if action is not on them is interesting. I'd be curious to see if it would result in improved play.