The reason I think stalemate makes the game more interesting is it gives an out to the losing player.

Suppose a game where black just lost their last piece (outside of the king). With the stalemate rule, white still needs to be careful and skillful in executing their checkmate to actually win the game. Giving black some way even in a losing situation to get theirself into a better outcome than a flat out loss. It allows more opportunities for games to reach their natural conclusion in a way that still allows both players to do something to get a better outcome.

And I don't know about you, but I think a game that could end up as a draw right up until the end is a more interesting game to play and much more entertaining to watch than watching 30 turns of white maintaining an early all the way to the midgame when black decides it's just a waste of time at this point and resigns.

It shouldn't. The whole point of a check mate is that the king will inevitably be captured. If you're in no position to capture the king, then you can't win.