Of course not, because then anyone in a zugzwang position would just say pass, chess would lose 75% of its logic and strategy and endgame complexities. And the result would be the same anyway. If here, black said "pass"
White would pass too because if he moves again he losses, so it would end in a draw anyway.
Checkmate would still be the way to win the game if you were allowed to capture the king. Checkmate by definition means the king would be captured next move. The question is whether stalemate, also meets this criteria of the king being captured next move. It doesn't, as stalemate doesn't mean the king will be captured next move, it just means the player can't move. It could mean the king could be captured next move, but it also could not, as the examples show.