So the problem is not single parent households at all.
The problem is single parent households where the single parent in question is incompetent to provide for children (shelter, safety, food, and spirituality).
It is our national policy of facilitating incompetent single parenthood that is the problem.Under no circumstances should the state, using parens patriae as justification, place a child in the care and custody of an adult who cannot make a living sufficient to support self and child(ren).
We never, never, never allowed this prior to the past thirty years. It has been a horrible mistake, and should be reversed immediately.
All of the other pathological consequences you have been discussing flow from this cosmically bad policy.
AFDC was to support single mom households to allow dependent children to stay with her. Dependency was caused by the father's death, imprisonment, or abandonment. It was then assumed she could not work and support a child at the same time. Times have changed. A father is expected to be a provider, hence, no government assistance programs for him.
Parens patrae powers enable the state to take charge of abused, neglected, dependent children(dependent on the state for their support). Until the father is shown to be unfit, the state has no power to interfere. The state can't award custody to someone other than the father without taking custody from him (his by right). This is no insignificant interference with his natural right, requiring some sort of parental termination hearing, not like weighing "best interest criteria", reserved for placement of children who are already wards of the state.
If the father has abandoned them, and could have supported them, he should pay. If mom won't marry the man, she should surrender custody to the father or encumber her own family with the debt. The father has no obligation where he is denied custody.
What do you think of the concept of presumptive joint-custody, in which absent abuse, both parents retain custody?