Without the stalemate rule then there's not much any point in continuing to play once you're in a poor enough position. You'd literally be wasting both players times waiting in vain for your opponent to irrecoverably blunder hard enough to turn the match, which may be impossible if you're out of enough pieces.

With the stalemate rule although you may no longer be able to win, you can still do something so your final outcome is better than a loss. The losing player still has a reason to keep playing. The game is overall more interesting to play and watch by having the stalemate rule than it would be without.