To the OP: Yes...and No.
Every situation is different. There are women who simply don't know who the father is. In extreme cases they may not have so much as a name.
There are women who know but just don't want him around, for reasons that run the gamut from them not wanting to share control of the child to the father's issues they believe would be harmful to the child if exposed to them. In that situation the father can press his rights through use of the courts and putative father registries, though. He always has recourse.
Then there are the men who abandon the mother and child and it's his choice not to have anything to do with the child. These men whether they're on a birth certificate or not aren't going to be involved and will fight taking responsibility.
Then there's the other extreme, the women who don't want or cannot supply a father's name on a birth certificate because it was a situation of incest, rape or abuse.
There are so many different situations out there it's difficult to come up with a one size fits all rule that addresses every one of them in a just manner. Which is why family laws are so complicated and the result (ideally at least) is always an individual determination. I would hesitate to apply either a one size fits all rule in this situation or allow an overworked, underpaid, cynical and most likely highly unqualified caseworker to make any kind of individual determination.
How can you not know who the father is? You're screwing so many people you can't keep track? Worse yet, I'll reproduce with you but I don't want you around?
Who are these people?
Yep, that about covers some of the situations. But many, many others? No.
There are men who ask not to be named, who don't want or care about rights and the women oblige them for a multitude of reasons. There are private contractual agreements between parties that down the road are broken and cannot be enforced in court without money she doesn't have. There are women who leave an abuser when they become pregnant and fear for their safety and that of their child if he is notified of her location, let alone the child. It's not that simple, there are as many fact patterns as there are individuals.
If a man is serious about wanting to be a parent, he can pursue his rights (and take on his responsibility) whether she puts him on a birth certificate or not. That's what putative father registries are for, and it can be done at any time.
As I said in the post to which you replied, it runs the gamut. Some men get screwed, some women get screwed, but the child is ultimately the one being punished. You have some serious constitutional issues here as well, but in the end it boils down to common sense and what's in that child's interests.