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.
While agreeing with you entirely that legitimaacy must be re-emphasized, I still say you have failed to show how your solution (abolishing financial liability for unwed fathers) honors the natural right of every child to his father's support and provision.
If only married fathers have this responsibility, while unmarried fathers can (in your scheme) get off scot-free, you further decrease the likelihood that men will seek marriage.
Women (plural) in general need to re-assert the sexual rule: no matrimony, no sex. OK. But in each and every case where a man and a woman have produced a child by their sexual union, both the man and the woman must be called to responsibility. Otherwise you have only the woman being treated as a responsible adult, and the man being treated as somthing less than an adult: either a mental defective, a rutting animal, or a perpetual lad.
I know a cute little young lady (now 30-something) who has 3 sons from 3 different fathers. She gets $1,500 a month from each of them. She doesn't work, she gets all the WIC support she wants, her house is a disaster, and her kids are snotty little undisclipled feral animals who will likely die in prison.
Plus, she drinks like a fish, she's slept with every guy under 45 in town (except me, yikes!!!), she literally exudes unhappiness, and ironically, she complains that she can't find a "nice guy" anywhere.
Not surprisingly she thought Bill Clinton was the greatest man on earth until Obama came along.
Oh yeah, she's also from California.