lampooning the other sides' argument.
For my part, it's less about the overall strategy and more about I think there should be more of a penalty to removing your pitcher from the game. In the NL, in most cases removing a pitcher also means losing a position player off the bench. I like having that as a balance against "unlimited" pitching changes - especially with rosters being expanded to 26. That's also why I like the rule they are adding that requires a new pitcher to pitch to at least 2 (or is it 3?) batters. I like the game better when there are fewer pitching changes.
That's also another part of the argument for me... I dislike specialization. I like players that play multiple positions and can be used in a variety of ways. I don't like how the game - especially the bullpen - has become so specialized. I like guys who can pitch effectively against both lefties and righties. And I like guys who can both hit and field effectively. Too often a guy is the DH because he's a liability in the field. That's the biggest turnoff for the DH for me.
I'd be all for the DH if it was used more as a way to give guys a day off from the field from time to time. I don't like how in the NL if you want to give a player the day off, he basically does nothing for most of the game. I hate it when you tune in to a game and find out that 3 starters are being given the day off that day. Sure, they need time off - but the DH could be used to stagger that a bit. Instead of just sitting out 1 game a week, you get a turn at DH one week and then sit out completely the next week.
Now I'm just rambling, but tldr: it isn't all about the strategy, although I don't like it when DH proponents try to minimize that aspect of it.