I think stalemate is an important part of the game because the opponent can force himself into stalemate. You consider this folly on his part, but I see it as skill. The king is not dominated, the winning player should have had the skill to finish him off.
If the losing player is "losing", he still has the chance of drawing by stalemate. ie he still has a chance to show is skill and is not completely humiliated.
Jacks and hopscotch are also viable options for leetle guuuuurls.