Strictly from the state's point of view, a paternity test means jack squat.
Sperm doesn't contain some kind of magical tell-all script. All it can say is "yeah, it's your kid" or "no, it isn't." It cannot find the father (I think you're thinking rape kits, which when fiiinally tested are very useful in finding serial rapists), and it cannot tell everything ever about a family's medical history. That'd have to come from the man himself.
mandatory paternity testing puts everyone in that doubtful state from the get-go. It may be popular on Reddit, but people living real lives in real relationships can see how the perspective of mandatory lab verification for morality's sake is a bit unhealthy. Paternity testing should be an opt-in right... isn't it already, though?
If the worst case scenario does happen, and the father isn't THE father, he doesn't sign the birth certificate. Yay, he's free*. But that child still is likely to have needs that two parents must work to fulfill. The child will likely need welfare, burdening the taxpayers, and it's not in the State's interest to facilitate that at the cost of the population's wallet.
A father finding out that his kid isn't his can petition to end his support responsibilities, but the child still needs two sources of support. A sperm test doesn't identify some third party and point the finger, is only says y/n to one applicant. So unless there's a way and a reason to more aggressively rope men into paying for children they had out of wedlock (or in someone else's marriage, or with another single woman), again, it's against the state's interest to leave children without support.
*unless he continues to hold out the child as his own- acts like a dad, in which case he would likely be liable for child support down the line.
I get it. The whole punish bad women for having sex thing is a huge motivator here. But ultimately, what people like to forget is support is the right of the child, and the state, having an already overtaxed and sometimes downright abusive foster care system, is reasonably loath to burden it further, just because couples don't know what the fuck's up with their own partner...