The only thing I can imagine working is for a birth certificate when listing father, would have a selection of “presumed” or “verified”, where verified would have a doctor sign off. It should be optional. But it would be nice if a paternity test was presented as an option automatically, without needing suspicion or accusation.
As it only requires saliva, any father can test their child’s DNA. Babies drool a lot. It may cost a few hundred dollars to do so. I suppose that makes it even less trusting to do it in secret than just having it as standard medical procedure.