But that happens anyway - any parent who is living in squalor and incapable of caring for a baby is subject to being investigated by social services, and in extremis, having their parental responsibility severed through adoption.
But the point is everyone here is working from the basis that the birth mother is the only person with parental responsibility, and that if she does not wish to care for the child, it should then fall to social services to decide who should raise the child.
That's not how it works. Provided the father is listed on the birth certificate, he has parental responsibility. If the mother does not wish to raise the child, but the father does, that is what will happen. It is not a case that social workers will decide first whether the father is suitable.
Of course social workers can get involved if they have reason to suspect neglect, abuse etc, as they would with any parent. But the starting position is that the father will have parental responsibility and the same (very high) thresholds for removing it apply as they do in any other situation.