Other than the NL and its minor league affiliates, how many leagues / levels play without a DH? Is there a DH in high school now? International uses a DH, I'm pretty sure college does.
Personally, I think a DH makes the game more of a players' game than a manager's game. By taking the pitcher out of the offense, you do lose the one out bunt, the double switch, the keeping a pitcher in too long because he is due up next inning, and the save your better reliever because you don't want to do a double switch and the pitcher is up next inning.
I'd give up that "strategy" for the chance to see a pitcher gut out extra innings when he gets shelled in the first in order to save the bullpen, seeing my best reliever for the situation come in, fewer automatic outs, and more skill at bat and in the field that you get out of regulars.