> Sorry, when I was talking about the stalemate rule, I meant the rule that declares that the game is a draw, if the only move you can make is putting yourself into check.

There is a Chess game that does this. It's Chinese Chess (which is just as old as Western Chess, and has the same origin in India). In Chinese Chess, the objective is to capture (or as the Chinese say, "eat") the king. There is no prohibition against putting your King in check. It works just fine, unless the only pieces left are Kings. The main problem is players doing checks forever or chasing pieces around forever: the rule is that if you do that you lose.

https://en.wikipedia.org/wiki/Xiangqi

Now stalemate by repeated back-and-forth is a different matter. I would favor some variant of Go's Ko rule, which is that you may not make a move if the result would be a board that looked exactly the same as it had in the past.