Wouldn't that be a further denial of a child's natural right to receive support and provision from his father? And a further encouragement for males to disconnect from adult obligations? Your solution seems one more step in the wrong direction. The desirable thing would be to require both women and men to be more responsible, not less.
No.
It would make legitimate, married fathers more important.
Hey, if you want more illegitimacy, subsidize it. Tell women if they can get themselves knocked up by a guy, they can collect a paycheck from him every month. If she gets knocked up by three or four guys she can retire.
Women who don't bother to get married before getting pregnant ought not to be able to avail themselves of the advantages of marriage.
When it comes to sex, women are in charge. They decide if they want to have sex with a man. What the man wants is irrelevant.
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.