I do not agree. Consider these payoffs for the same game: ...

Different payoffs imply a different game. But even in this different game, the simultaneous move version would be equivalent. With regards to choosing between X and Y, the existence of choice A still doesn't matter, because if player 1 chose A X and Y have the same payoff. The only difference is how much player 2 knows about what player 1 did, and therefore how much player 2 knows about the payoff he can expect. But that doesn't affect his strategy or the payoff that he gets in the end.