I've heard the slippery slope argument about "why just 1 DH, why not two separate squads of David Ortizes and Brendan Ryans?" I'd suggest there's a statistically-neutral reason for having one DH and not more. It is basically the principle of treating like things the same way and different things differently.
There is a huge (HUGE!) qualitative difference between pitcher's offense and other hitters' offense. By declining OPS, here are the wOBA and OPS of each position (using scoring notation):
OPS wOBA
3 - .780 .336
9 - .758 .328
5 - .738 .319
8 - .734 .320
7 - .730 .317
4 - .706 .307
6 - .682 .297
2 - .678 .296
1 - .326 .147
(source - fangraphs league stats table - all leagues)
Among position players, the greatest gaps are between left fielders, 2d basemen, and SS - .024 points of OPS, and .010 of wOBA. In contrast, the gap between the worst hitting position and pitchers is more than double the pitchers' OPS and wOBA (e.g., more than .350 in OPS and nearly .150 in wOBA). The difference between the very best position for hitting (first base) and the worst position for hitting (catcher) is about .100 points on OPS and .040 points on wOBA. Basically, pitchers are playing a different game when they appear at the plate then any other position. While the other positions are balancing offensive and defensive contributions, offensive contribution from the pitcher is nearly a random afterthought. They don't belong with a bat in their hands.