Child support is not proof of paternity. I know many young men hung with supporting someone else’s kid. In many states you only have to be married to the mother when the kid is born and the kid is yours regardless of paternity. Or the mother can accuse you of fatherhood and if you don’t fight the kid can become legally yours.
The best thing to do is eliminate all forms of child support. If a women doesn’t want a child it is within her power to never have one. Why should anyone ever have to pay for someone else’s voluntary activity?
Not supporting your child is despicable but it is something best left to families, churches, and local communities.
Your scheme won’t work because women can still vote.