To select a move, one considers the situation from their opponent's perspective, the other side of the chessboard. It's a more effective thought experiment when one knows who their opponent is, and why they might make non-optimal choices.
The fallacy of the concept is that, in chess, both players have the same goal (capturing the opponent's king). In attempting to apply the concept to Beatrice, both Kyrie and Battler (in and out of the game world) assume Beatrice has the same goals as themselves (obtaining the Ushiromiya inheiritance, or forcing the other party to accept their version of the Rokkenjima events). Unable to understand her true motives, both fail to gain any understanding from turning the board around.