Laws vary from place to place to place.
This. A thousand times this. Let's never forget this.
The loophole of marriage being a prerequisite for being recognized as father is pretty damn bad one. Marriage shouldn't matter at all.
And how do you adequately measure contribution, anyway? What if the father wanted
to contribute, but was repeatedly told to fuck off and never show his face again? Or the mother was only willing to accept money, which might not even have reached the child? (Let's be honest: monetary payments do not guarantee that the child ever gets anything from it, anyway - there was a nasty story of a mother throwing out the father, then demanding child support, getting it, and drinking it all away and nearly letting her child die not too long ago, for instance...) Or the mother never even informed him of the fact that she carried a pregnancy he did not know existed to term despite him very much wanting a child of his own and being well capable of caring for one? And again, there are cases where man will be struck with child support payments "out of blue" - it may not be terribly common, but it happens.
Whether is it taking over the rights or surrendering them, in any case it is a bloody mess.
Unless you have those people under constant surveillance, it often boils down to word against word, and both males and females are equally likely to lie or speak truth respectively.