If there was no check this would fundamentally change the game, because stalemate would no longer be a draw but rather a loss for the stalemated side, because he would be obliged to move his king into check and have it taken. This would have big implications for endgame theory - i.e. a lot of it would have to be thrown out and redone from scratch, such as K+P vs K becoming a win in almost all positions, rather than just some.
Why is it illegal to step into check?
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Why not just let the opponent capture the king? Why do we have to announce check, either? I say we take the training wheels off.
Because if the king were killed, there couldn't be another chess game..
But I think it has to do with the idea of chess. It is meant to be a civilized version of war. Allowing for sportsmanship. It reminds me of the fencing scenes from old movies, where one guy strikes the sword out of the other's hand. Instead of proceeding to thrust his sword into the defenseless mans stomache, he just aims the tip of his sword at him to signify defeat. Of course, in real war you would probably want to kill the king but on a wooden board you don't need to worry about that.