how different would chess be if you could capture the king, removing check and checkmate entirely?

Would the game end, would it continue, or would reality just collapse?
(sorry if this should be in off-topic)
Depends on exactly what you mean. If you mean changing the goal to capture the king, would you allow moves that expose the king to attack?
If so, that will add people trying to trip up their opponent, especially in blitz or time scrambles.
Overall, it changes the dynamic of the game, so that many endgames that are draws today, would become wins. That can change decisions on some lines, earlier in the game, to try to prevent those kinds of positions from being reached, or potentially trying to reach them for the side that would get a winning game.
It would make no difference at all.
Stalemate would be a win, but that makes no difference: chess stays a draw just the same.
https://arxiv.org/pdf/2009.04374.pdf Figure 2b

Would the game end, would it continue, or would reality just collapse?
In some traditions, chess ends with capture of the king. It does make checkmate and stalemate easier to understand if you end with king capture. So it would change nothing.
Sometimes beginners struggle with a checkmate like this. Ending on king capture makes it much easier to understand.
Would the game end, would it continue, or would reality just collapse?
(sorry if this should be in off-topic)