Stalemate has been treated as a draw for 200 years, but that doesn't make it right.
I agree with GM Larry Kaufman, who argues that Stalemate is the ultimate form of zugzwang, and the player making the move that creates stalemate should win.
On the other hand, it taught one of the high school chess players I coached a valuable lesson. He was stuck around a 900 rating for a couple years, then stopped playing cheap traps and his rating rose to around 1200. One match, playing a weak, unrated player, he decided to Queen every Pawn he had before applying checkmate, humiliating and torturing his opponent with each move. BUT after Queening his last Pawn he had five or six Queens on the board and his opponent had no legal move: DRAW by Stalemate! His face turned green! He never acted so dispicably again.
Why not just have it the player missed it, his king is dead, game over.
Not necessarily. Both players might not see it and the match simply continues.
But I agree with you, I would love to try that out and a chess site like chess.com is an ideal place to do it, simply create a new page were people can try out the "new" chess rules.
That is:
1. king can move to all free squares, like all other pieces can
2. match is won when the king is taken (or when opponent resignes or time runs out)