The child is already there. Neither of them can un-have the child. Having a child is both the parents accepting a responsibility to provide for the child in the future, regardless of whether they stay together.

Obviously, the child support should be formed in such a way that as long as the children lives with both parents (either because they are together or because the child lives 50/50 in separate locations), there is no need for transfer of funds. If one parent wants to (or is forced to, e.g. after abuse) give up caring for the children (so the child lives e.g. 100% with the mother and 0% with the father) only then should that parent have to pay for child support.

I'm not sure whether that is the way child support works in the US but it's pretty much the only reasonable way I can imagine.