Would many american woman father to terminate parental rights and get no child support in this case?

Would many American mothers do this?
Would many American mothers allow the father to terminate parental rights and get no child support, if he files a $30,000 lawsuit to become the primary custodian, and you many eventually have to pay him child support? The trial court last up to two years or more.

Are there many American fathers that after a divorce will terminate his parental rights over his children?

Are there many American fathers that after a divorce will terminate his parental rights over his children, when the mother asked him to do so? In return the father does not have to pay child support. The father knows that the mother never wants him to see his children again, and that is the only way the children will not suffer from a divorce.