I don’t think anyone here is saying that society will fall apart or anything, and you are right that costs alone should not be a reason to dismiss the idea out of hand.
But how do you propose to handle the practical application of such a law? Do you propose to DNA test every child born, comparing the sample just to whomever Mom claims is Dad, or would it have to be compared to every person Mom thinks might be Dad, every person who claims to have had sex with Mom within the past 9 months? And what about when Mom says, “I don’t know”, would those children be exempt from the law?
Then what do you do when you know whose DNA contributed? Put them on the birth certificate? Hold them responsible for child support? Hold them responsible to raise and love the child?
What about a child conceived using a donor egg, donor sperm and a surrogate? Far-fetched I know, but bear with me here… Would the egg donor become Mom and the sperm Donor be Dad, or would the surrogate be Mom and the donor be Dad? Who would be the legal parents responsible for supporting, and raising the child?
It isn’t simply a matter of costs, but there are too many scenarios in which the proposed law (Paternity testing at birth) wouldn’t work. If children in unusual circumstances were exempt then the law would be unfair, it just isn’t practical.
Not to mention that this would (or could) create a national database where every child’s DNA was a matter of record and in several generations every person born in the country would have their DNA on file. That’s rather invasive and I believe would be a violation of other rights that we have.