Having the DH universalized would make managing too easy. There would be no double switching or taking a pitcher out in the 6th inning when he still has good stuff but there are men in scoring position with 2 outs and the possible runs are more important. Or anything else resembling actual strategy.
I actually am pro-DH. Yeah, that's surprising, coming from a fan of an American League team, eh?
The argument about strategy falls flat for me. Most of the pinch hit/double switch moves i see in NL games are as predictable as anything else -- if a pitcher is a poor hitter, and its a close game, he'll be pinch hit for, unless the roster is deep enough to pull the switch, etc. I would argue that the DH rule simply alters the strategy. While a manager no longer has to worry about pinch hitting for the pitcher, he has to watch more carefully, and pull his pitcher based solely on how he's throwing (at least 75% of the time, maybe more, if a late game situation calls for a pinch hitter in the NL, the manager WILL pinch hit for the pitcher, regardless of how well the pitcher is throwing. With almost any pitcher in the NL, if you're a run down, a guy on base late in a close game, the manager will pinch hit). Indeed, early on in the history of the DH in the AL, Billy Martin, then manager of the A's, paid little attention to his starting pitchers, and burned them out, overusing his starting staff. Its a much more subtle strategy. The other thing the DH offers is instead of a poor hitting pitcher or a mediocre bench player coming to bat, you usually have a well established star-level professional hitter in that slot. For all the hoopla about pitchers occasionally coming through with a hit, or the crafty bench player getting the key base hit, most of the time, this doesn't happen. I'd rather have the excitement of seeing a Frank Thomas or a David Ortiz facing your bullpen ace in that situation.
Now don't get me wrong -- I like the game when the pitcher bats too. Its just a different strategy, and I have become accustomed to the DH rule as a fan of the White Sox. There is sort of a snobby arrogance amongst the national league fans I know -- hey, the game is just better, more pure, in the NL. Pish posh! Its just different. And, by the way, only two professional leagues in all the world DON'T use the DH -- the NL, and one of the top leagues in Japan. (Of course, most of the NL fans I know are Cubs fans, so I guess such comments must be considered in light of the lack of logic, brain power, and objectivity such fans possess).
In fact, if eliminating the DH would make both leagues rules the same, I would go for that. There might be a way to get around the players' unions objections. If baseball were to expand again and add two teams (making a total of 32) and potentially expand rosters by one or two players, the increase in the ability for aging players (i.e. Frank Thomas, Jim Thome, etc) to compete for jobs would be there, and the DH could be effectively eliminated at the same time the last two teams are added.