Feminists have thought this out; they are not opposed to limited female privileges such as the exemption from combat duty, or as you point out, the right to child support. Both of these position ultimately derive from the natural law and are perfectly rational forms of discrimination,making them constitutionally legal and protected.
Natural law does not provide for the exemption of women in the military, in combat positions, or in combat itself. Warfare and combat as defined and viewed by the Church takes it’s roots in the moral right and responsibility everyone has to defend themselves or others. Women have no less a right and responsibility then men. Then of course there is the issue of St. Joan of Arc (either she wasn’t really a saint [invalid since the Church has declared her such] and wasn’t acting under the direct orders of God [invalid again]; or God violated natural law [invalid argument]).
Men, by and large, pay for their children because they are “stepping up to the plate”. Other simply ignore the obligation. The case you’ve describe is already effectively possible; by societal privilege, men can simply dodge any financial responsibility for their children. Court orders you pay child support? Simply don’t…

A minority of men may choose to pay child support based solely off the risk of contempt of court charges, but such charges are so rare as to be effectively meaningless.
Your example has nothing to do with the case I described earlier. In my case the man is legally protected from having any responsibility for the child. In your case the man simply ignores his legal obligations and the law itself. You’re trying to argue the equivalent of lifting/increasing the speed limit won’t have any legal/social impact since everyone already breaks the law and speeds anyway.
This failure of masculinity is the primary driver of abortion in the world. If fathers stood up and were willing to protect and provide for there children, far fewer would be aborted. Mothers would not feel so desperate as to allow a ‘doctor’ to stab or poison her child rather than risk watching it grow up in poverty, and by extension, put her in poverty.
I’d love to see where you get this idea from. It’s not the woman’s fault she goes out and gets an abortion. It’s, of course, the bad old evil man’s fault. Please.
So to suggest that legalized abortion “logically” means that we must also legalize paternal abdication is a trite argument.
Paternal abandonment is already the norm; making it
de jure would have the negligible added effect of appeasing these loser’s consciences slightly. Abortion is sought after because of this very problem, and arguing that it is only logical to make the problem slightly worse is simply not an effective argument.
You can’t abdicate what you don’t have. Perhaps you should reread what my OP is actually stating. As for trite, someone putting for your position on why the majority of women get abortions shouldn’t really accuse others of having a trite argument.