So the children would be suing a parent for being born?

Fair question. It's unlikely to succeed (for one, wrongful birth causes of action are barred in a number of states, and the damages aren't apparent for a very long time) but there ought to be liability when one secretly fathers so many kids as to move the sibling-parentage foreseeability needle away from zero.

More likely to succeed are child support-kinds of actions and claims against the physician's estate by all of his unackd issue if he dies intestate or fails to call out and disclaim every one of them expressly in his will.

One thing is certain: This fact set has the makings of a really good bar exam essay question.

That seems like a legal can of worms and a half to open up.