If in chess, pieces have the option instead to not move, then how would the game change? I have looked through a few positions and I see that zugzwang doesn't work at all. What else would be changed?
Moves in chess are powerful. As any player or even engine will tell you, a move has material significance (Stockfish awards it a fifth of a pawn, visible when no moves have been played).
There are some mechanics of chess, like stalemates and, as you mentioned, zugzwang, but for openings and middlegames , it really wouldn't change much.
As an example, beginners often are unable to formulate a plan and thus waste moves, this is analogous to passing your turn and often results in a loss.
Endgames are another story entirely. King and pawn endgames rely on zugzwangs (a common example is where one side tries to infiltrate the other's pawn chain) and opposition will not be a thing anymore, thus drawing most games without a significant advantage. To put into perspective the severity of the damage, even King vs King and Rook will be drawn (because of no opposition).
It may also slightly increase wins in positions where one side plays for a stalemate, but in my opinion, the game would overall take a turn for the worse, as it mostly affects only the endgame and can ruin a hard fought game with a undeserved draw.