It's kind of an illusion when you think about it. "Whose turn it is" is an inseparable part of the game state. If any move makes the game state worse this turn, then the game state was already bad before this turn.

You can infer the game state from way before a zugzwang is played out on the board, and if you're on the losing side of the eventual zugzwang, it's normal to resign.

But if you were allowed to pass your turn, and both players see the draw coming because of a forced repetition, they'll just call it a draw before it even plays out. So the game would play out differently from the same position, if that rule existed. Essentially changing the way you would evaluate any given position.

It's not necessarily an illusion. If chess is solved and it turns out white wins with perfect play, black's first move is zugzwang.