In my view, this will never, ever happen for one simple reason - if the father of a child isn't responsible for the financial wellbeing of his child, the child will, in all likelihood, become a financial burden on the state. The state will never pass a law that makes them the child's supporter by default. And as a taxpayer, you should never support any law that makes the state "daddy".
Eh... this is making a lot of assumptions.
First of all, that child support payments prevent this. That isn't true.
For women who truly cannot support their children on their own, child support is usually a drop in the bucket. It doesn't even begin to cover the true expense. Plenty of women get both child support and assistance.
Also, a woman who is so poor she cannot afford her children probably didn't have them with a rich man. Oftentimes, he can't afford the payments. So a lot of the time, he doesn't pay at all. It's relatively easy to get out of child support payments. Even if he doesn't and his wages are garnished, that can effectively put him on the dole himself, in addition to doing almost nothing to help the woman since his wages are so small to begin with. You will likely still wind up with at least one person on welfare in this situation -- which applies to an awful lot of people.
Second, the reality is that it is the child of the one who accepts responsibility for it. A woman who puts a child up for adoption is not responsible for its well-being despite the fact that she is responsible for its existence. Why should a man, who has basically done the same thing, be forced to pay where a woman isn't?
Your argument would apply to a bio mother who put her child up for adoption even MORE than it applies to a man who wrote away his rights: a child put for up adoption will almost certainly require public support. By your logic, adoption shouldn't be allowed due to the near-certainty of the child requiring public support.
For that matter, it could apply to public schooling. Why do we pay for the education of a child you made?
Also, the fact that this conversation always goes in one direction: money from the man to the woman. Women can work these days, and there are single fathers too. Why do we always talk about this in the subliminal context of the "weaker sex"?
This gets really screwy and hypocritical really fast.
But the reality is this. First, that child support does not cure ills. In some instances, it simply makes more. Second, that we live in a society that is sufficiently complex that none of us can be said to be truly independent. And third, that even if neither of those were the case, it somehow justifies forcing a man to spend his life and income a certain way for something he had no say in.