Is there a duty of care for wrongful birth?

Some bad assumptions here. There is no such thing as a wrongful birth, at least by what is implied in the sub statement. Any birth results in a child unable to care for itself. The duty is of the parents to provide for it. If the state determines the parents are irresponsible, then the state assumes the duty. Otherwise, the parents may try to get another to assume the duty of raising the child through adoption. But, regardless of the law, the logical connection between the act of reproduction and a birth leads directing to the duty of seeing the child is cared for.

Playing philosophical games with child care responsibility leaves some adults confused on what is right and wrong and thus results in unnecessary problems for children and society.

It's called Child support for the reason of your responsibility. If you don't want that then have your parent rights wiped from you.

yes

you play you pay