On the other hand, bots are starting to beat professionals at (thousands of hands of repeated) Poker, so I think we can't say that imperfect information is something that's especially intractable for maching learning algorithms.
Exactly. And poker is, again, turn-based with a single move to be made. And it may not be "perfect information," but it again is a game of static board state where there is probably a statistically optimal move for a given state provided you have memory of the other players' previous moves so far in a round.