Examine your premises more closely. If a woman does not have the ability to use government force to get support from the man, then her only option is to encourage the father to emotionally bond with the child, and her, well enough that he is willing to VOLUNTARILY support the kid.
It would also act as an incentive for her to make better choices in who she has sex with.
It is not power which corrupts -- it is the absence of consequences which corrupts.
In has heretofore always been in women's best interest to refrain from sex until real bonding commitment are reasonably to be expected, leading to the more circumspect behavior you and I both approve; but the availability of abortion has changed that whole dynamic.
Ultimately, nonmarital sex is still self-defeating from a woman's perspective (men's, too, but it's more obviously and dramatically so for women) but the path toward correcting this situation must aim at the re-assertion and reinforcement of both men's and women's sense of responsibility, not going for some fragmentary solution which demands women's performance on their adult obligations, and leaves puerile males to do as they please --- as if we expected them to be fly-by-night inseminators.
I respect men enough to demand strict liability from them (as well as from women) for their sexual behavior.