A way to improve chess: get rid of stalemates

Ever since I started playing chess in the 90's, I've not been cool with stalemates. I understand that it's an established rule and not many players want to change it. However, I do feel it's illogical and Chess would benefit from changing the rule.

The reason I think it's illogical is that one player completely dominates the other player but stalemate ends the game in a draw. It just doesn't feel fair. Think about a war. One army manages to kill or completely immobilize the enemy army and the chief commander is surrounded and unable to escape. It's the end, but suddenly the dominated commander claims: "I can't do anything, let's call it a tie" and so they agree it's a tie and everyone goes home. That's not how battles in war work.

The reason I think changing the stalemate rule would benefit chess: Yes, it can feel good to survive an overwhelming onslaught if it ends in a stalemate, but even if I've been able to survive in this manner, it just haven't felt fair. On the other hand if you stalemate the opponent in an overwhelmingly superior position, it's the most painful feeling and I've had feelings of hating the game because of this dumb rule. Also a good percentage of the draws in Chess are the result of stalemates or inevitable stalemates looming. This would be a way to address the dreadful drawishness of the game and make it more exciting. I personally don't have issues with the result of draw itself, but maybe there's way too many of them in the top level.

The way I would change the stalemate rule: make moving the king into check or leaving king into check legal. Also make capturing the king legal. Capturing the king ends the game and the player whose king was captured loses. This would virtually eliminate stalemates. Or just count stalemate a win for the player who stalemates the opponent.