Your claim about the logical outcome of libertarian thought is wrong, because it fails to consider a key fact: the child is there because of the parents' choice. They chose to bring the child into the world. Or, if they adopted the child, they chose to do that. And when you make that choice, you accept the obligations that come with it. There is nothing in libertarianism that prevents people from making choices that come with obligations that they must accept. The only thing libertarianism would be against would be forcing people to raise children they did not bring into the world or did not adopt.

There are edge cases, such as a woman becoming pregnant against her will (rape, for example); but those are edge cases and don't change the basic principle.