What problems would it cause?
Stats say that 1/3 of married women have had an affair, and about 1/2 of married men have had affairs - no doubt with some married women. I’ve heard that 10% or so of people’s fathers isn’t the guy listed on the birth certificate, which pretty much jives with that 1/3 and 1/2 figure of marries diddling around.
My guess is that DNA testing at the hospital at an otherwise happy event would cause even more marriages to break up (even if the dalliance is in the past, or a onetime thing) than usual, as well as fits of violence.
I don’t think much good would come from this, sorry.