I was not talking about the rule that handles games that do not progress. (Though I would perhaps replace that rule with one that just forbids repeating board states, but that was besides the point here.)
So to make chess more elegant in how the rules are formulated (but probably not a better game, mind you), I would suggest removing all special rules around check and checkmate. Just replace it with the rule that you lose when your king gets captured (and obviously, keep Zugzwang).
Sorry, I was not clear that I also wanted to remove special handling of check and checkmate.
(Btw, this feels very similar to the rules against self-sacrifice in Go.)