The paternity issue should be easy to overcome with modern technology. There's really no reason the state shouldn't require a paternity test to ensure the accuracy of the state issued birth certificate.
Some states go the other way - if you (as the father, maybe the mother but that's pretty easy to verify I hear) sign the birth certificate you are the father (Maury) for all legal purposes even if you're not - wether knowingly or not.
Yes, most states see it that way. But you could still make them the legal father through adoption (like with step parents) without providing inaccurate information on the birth certificate.
If it's fraud it's on the part of the mother (usually) - the state just wants a clean database to avoid complications; it doesn't actually care about truth.
It's similar to how the justice system doesn't really care about justice, it cares about detailed and systematic applications of the laws, which sometimes coincides.
"the state just wants a clean database to avoid complications; it doesn't actually care about truth."
What it actually cares about is not having to pay CHIP, etc. Having a man who isn't the biological father paying for stuff means the state doesn't have to pay for services. They don't care at all beyond that.