I'm dubious that this lacks societal benefit overall. Even if we buy that breaking up the sexually abusive family is't the best thing (already dubious), there's strong second order effects of this disincentiving abuse.
Automatic genetic testing also has other advantages of preventing future paternity disputes.
Honestly, this would be an interesting thing for a single state to run an experiment with and see how behaviors change.
While society would no doubt benefit from outing abusers, teen births as a whole are a tiny portion of births. 10% of a small number is a small number.
Given women cheat at the same rates as men, blanket genetic testing would break up a significant amount of families. That's why some countries (France for example) ban paternity tests.
Automatic genetic testing also has other advantages of preventing future paternity disputes.
Honestly, this would be an interesting thing for a single state to run an experiment with and see how behaviors change.