I don't think it's illogical. Quite the opposite.
The point of checkmate is this: "your king is about to be taken, cannot be saved in any way, hence you lose".
Stalemate is not the same thing, because there would be no king capture in the next move. It is a consequence of two rules:
- To win, you *must* checkmate the opponent's king, as in "give an imparable check". This is the winning condition of chess.
- A player has to move on each turn, and cannot pass the move.
In a stalemate, one player cannot legally move, and the game cannot continue, but there is no winning condition met. A win without winning condition would be illogical. A draw is the only logical conclusion.
Logical or not logical, sometimes you have to accept the rules of the game. You could say that the rule for castling is illogical, or illogical that you can have several queens on the board after pawn promotion, but that's nothing you can do anything about.
One of the strength of the chess game is the fact that the rules have been the same for a couple of hundred years. The whole mass of theory have been accumulated through the years until the point where it is now, and it continues to develop. If you change the rules, you have to start over from scratch. For instance, if you change the stalemate rule, the entire endgame theory would have to be rewritten. Hundred of years of experience and theory would be worthless, and I doubt that the chess world would ever recover. Chess would be just one game among others, and it would be nothing special about it.