Is that true? I was unaware that PPO, SAC, DQN, Impala, MuZero/AlphaZero etc would all automatically Just Work™ for hidden information games. Straight MCTS-inspired algorithms seem like they'd fail for reasons discussed in the paper, and while PPO/Impala work reasonably well in DoTA2/SC2, it's not obvious they'd converge to perfect play.
If I remember correctly, the DeepMind x UCL RL Lecture Series proves the underlying Bellman equation in this video: https://www.youtube.com/watch?v=zSOMeug_i_M
As for "hidden information" games, I thought the trick was to concatenate the current state with all past states and treat that as the new state, thereby making it an MDP.
(History stacking may turn POMDPs into MDPs, but I don't know if they handle the specially adversarial nature of games like poker. That's quite different from stacking ALE frames.)
One intuitive way to see this is that a sequence of improving pure policies A < B < C < etc. will converge to optimal play in a perfect information game like chess, but not necessarily in an imperfect information game like rock/paper/scissors where Rock < Paper < Scissors < Rock, etc