As a practical matter, if the mother of a child, for whatever reason, doesn't inform the father that he fathered their child, then that man is out of luck. Then again, for some men, they might actually think that they lucked out -- no obligations, no responsibilities, a clean conscience, blissful ignorance, as far as they're concerned.
The potential father may or may not have the rights, that the OP described, in this century or the 24th century. But that is almost besides the point. I don't think the Federation is a nanny state, nor should it be. Should it be the business of the Federation, or any authority, to be invading anyone's privacy in this type of matter?
If a potential father wants to take the initiative, he is free to do so. If he finds that he has a case, then he can can go through the legal process if there is one for him to go through. Or he can simply try to contact the mother or child.
In Picard's situation, he could have put on his Dixon Hill persona and investigate (or try to contact) every women that he has been with to determine whether he might have fathered a child with them. Heck, he probably could get Data to do a search of all Federation data (no pun intended) bases to see, if the any of the women he had ever been intimate with, ever had a child near the time he was with them. I take it that Picard was never interested in doing so.
When Picard explained he would have been in his life had he known, Jason mentioned something like maybe that's not what she wanted. Miranda moved to a planet with a bunch of orphans living there. I am sure some of those kids probably had no clue who their father was either.
I got the impression that it was easy for Picard to say those words when he did. Would the Starfleet career oriented Picard really have committed himself to be in a child's life in any meaningful way, back in the day?
It was established that Picard is a character that is uncomfortable around young children. Picard seems like a man who might not mind having offspring, so long as they're already adults. Picard doesn't come across as someone who would be committed enough to raise a child from the very beginning, to go through the growing pains.
Picard seems like someone who would rather dispense with child rearing, leave that up to someone else while he flies around the galaxy; but then wouldn't mind having that child cozy up to him and have a relationship with them after that child had become an adult, when the child no longer requires the parent to go through the daily grinds of child rearing.