If you're in check and your opponent runs out of time to take your king (as stated by the proposed rule change) then, have you really won? No, you lost the game and in the opponent's time expiration you take the victory away from him.
This would be good to teach new players or younger players, but check mate the way it is, should stay.
Putting someone in check mate is the ultimate goal, and a sense of true accomplishment comes out when you win out of skill, not luck. Leave this rule be.
Do you have no morals, it was set up just take control and make the king feel bad not make him feel like crap
ahhahaahhaha