You missed one important point. If states allow named fathers to escape paternity via DNA testing, that leaves some children without a father who owes child support -- and then it becomes up to the state to assume financial responsibility, usually via welfare.
In a contest between whether the state assumes financial responsibility or some poor sap gets screwed, guess which one most legislators will choose.
I agree that the responsibility not exercised by the father generally falls to the state. And that is not an admirable outcome. But the specter of expanding the welfare state cannot be grounds for holding responsible men who are proved not to be the biological father. In that regard, it doesn't seem to me quite accurate to say DNA testing allows named fathers to "escape" paternity. Seems to me the attempted escapees are the known biological parents who fail to step up to the plate and raise their kids. But there's no doubt that most legislators will choose to "screw the poor sap" rather than assume financial responsibility, not least because three decades of feminist agitprop on these issues has pretty much poisoned the well for males as a group.