Probably because no one wants such an intimate and special day interrupted by a Government mandated medical test that takes three days? And I'm guessing with such a huge increase in the number of paternity tests the chances of mistakes being made would increase rather dramatically in the short term. But I mean you're free to ask the future (possible) mother of your child you want a paternity test in the maternity ward if it's something you're concerned about.
Yeah. In theory, I think there's a case to be made for verifying paternity. It does happen that people end up raising someone else's kid unknowingly, and that's not particularly fair. If there was a quick, easy, painless, accurate, and cheap way to make paternity testing an ordinary part of the process, I'd say go for it. Why not, after all?
However, my gut suspicion is that this is born less out of a desire to think ahead and help people out, and strikes me more as a "Men's rights!" thing. "It's not fair that women can trick us into raising some other guy's kid and get away with cheating! This is yet another issue us guys are drawing the short straw on, and I'm sick of it! Let's make paternity testing mandatory so all those bitches can't pull this crap on us!"
Of course, maybe I'm just too cynical and this idea arose when a legitimately swell, trusting guy later found out he was cuckolded.