It would function simply: If a player puts their king into check on their own move, if the opponent sees it, they can capture the king on their turn either by physically taking out the king or announcing checkmate on their turn. If the opponent doesn’t see it, the game continues as normal. Likewise, if the opponent puts the king into check, both players can respond on their turn if they notice it, or it’s simply a blundered opportunity.
The classical checkmate ending can be seen as an honorable ending, wherein the player effectively resigns/surrenders, or the player can be dishonorable by drawing out the game by making one more move. I understand the redundancy of this type of ending, but it would add more personable nuance to the game.
This idea can also be extended to stalemates: The only true stalemate would be one where a capturing of the king will not occur in a finite number of moves, e.g. move repetitions, or two sole kings on the board. A stalemate by trapping the king, ie the king has no alternative moves that wouldn’t put it into checkmate, should be a victory rather than a draw — the king would be forced to make a move into check, then the opponent would capture the king on their move if they notice the possibility.
A lot of chess variants do this, like Duck Chess and Drawback Chess. It’s especially useful if there’s a possibility of something outside the core chess rules (e.g. the duck or a player’s drawback) that could actually stop them from taking the king.
One extra detail to be aware of if you want to play this way: you should (or should at least consider) add in “castling en passant”, where a piece landing on the space a king left on the turn after it castles, or on the space the king passed through in the act of castling, also counts as capturing the king.