No it doesn’t.
If Alice has sex with Bob and Charles, and Bob’s sperm creates a baby, it doesn’t matter that Alice and Bob and Charles had an agreement that Charles would be the social father. Or rather, it only makes a difference IF NO ONE INVOLVED DISPUTES PATERNITY.
If Alice and Bob and Charles together agree that Charles will the social father no matter what, and then 9 months later Bob changes his mind and insists that he’s the father, he can demand a paternity test and will the the legal father of the child. Or, if Charles changes his mind and insists he’s not the father, he can demand a paternity test, and will not be the legal father of the child. Or Alice could change her mind, and and insist that Bob is the father and demand a paternity test that would make Bob the legal father.
However, if NO ONE DISPUTES PATERNITY, and Charles goes on the birth certificate as the legal father, and five years later changes his mind, a paternity test will not allow him to get out of his parental responsibilities, any more than a genetic test would let an adoptive parent out of their parental responsibilities.
Social or legal fatherhood does not depend entirely on genetic fatherhood. A bare assertion “I am the father” is enough. We don’t require any sort of evidence that the putative father ever had sex with the mother. It’s only in the event of disputed paternity that genetics becomes involved.
When my wife went into labor and gave birth, nobody at the hospital ever questioned me about whether I ever had sex with my wife, or if ever blew a load inside her. As far as they were concerned, whether I fucked my wife or not was irrelevant. I was there, she asserted I was the father, I asserted I was the father, and that was good enough for them. If tomorrow I decided I’m tired of this fatherhood crap and insist on a paternity test for my 8 year old daughter, and the test comes back that I’m not her genetic father, THAT MEANS NOTHING LEGALLY. Whether I ever fucked my wife or not, whether other men ejaculated sperm into her vagina or not, I’m the legal father of that child.
The “hey, you fucked her, so she gets to decide your the daddy” angle is nonsense anyway. It isn’t like fathers couldn’t deny paternity back in the days before DNA tests. They did all the time. It just wasn’t true that all a woman had to do was assert that a man had sex with her, and he was instantly the legal father. It’s nonsense.