AbigailII:
Draws are a fundamental part of the game of chess, and very common. Scoring a draw as a loss is a big change of the game. You might as well replace the rooks with diamond aces. It just doesn't make sense.

Giving both players 0 points in case of a draw doesn't equal "scoring a draw as a loss", and if the game is played as part as a match, it doesn't change the game at all. It simply means that draws aren't counted, and the match continues as if this game wasn't played. World championship matches have been played this way (but not with the doubling cube, obviously!).

As you pointed out yourself, when draws are scored as 0.5 there's an undesired side effect of the doubling cube: The leading player can double the effect of a draw, thereby bringing himself closer to winning the match. This can be remedied by not scoring draws at all, and I really can't see any serious drawbacks of this solution.

Btw., how do diamond aces move on a chessboard?