I think it should depend on the individual's circumstances. There are circumstances that are often created by the court that put the non-custodial parent at a severe disadvantage, and there can be circumstances beyond a person's control too. A love parent who is separated from the child/children he loves is already at a severe disadvantage because he's dealing with a grief that may last forever, and that he has to push aside and find a way to live with. It may be better if the non-custodial is certain that is children are absolutely fine and happy being separated from him, but most kids love both parents equally; so part of the grief a non-custodial parent can have is knowing is kids are feeling the grief/loss of being separated from him. Also, courts aren't particularly known for making sure the most loving or "best" parent gets custody. They're known for what makes sense and/or maintaining as much normalcy as possibly for the children.
A good parent will be sickened by the fact that s/he isn't able to financially contribute to his/her own child/children's well-being - and that's one more thing to feel grief about. It may not help that the non-custodial parent isn't likely to want to be open about the grief, because it can be human nature to have pride and not let one's "opponent" know that s/he has managed to hurt him and cause such grief by winning custody.
A big problem is the courts' pretense of "only caring about the children" because when children have two loving parents, and the court steps in and ties the hands of one of them; that's no "caring about the children" or about trying to minimize the sense of a "broken" family (as oppose to just a separated one). So, I don't think there's a one-size-fits-all approach to handling the difficult situation of the non-custodial parent's inability to pay support; and I really don't think there should ever be the automatic assumption that it is his unwillingness to support his children. There needs to be better protection of the rights and means to work of non-custodial parents. That's not saying there aren't "genuine deadbeat dads out there, but I think, much of the time, it's often more a case of beaten up dads. So, I think sorting out which fathers, exactly, a "genuine deadbeats" and which are, instead, "beaten up" (and physically, mentally and/or emotionally exhausted) fathers is the first thing the court system,and anyone else involved, ought to try to do. It's not just a parent's responsibility to support his child, it's his right. Protecting the rights of non-custodial parents by dumping the thinking that if someone doesn't live with the child he shouldn't get even temporary assistance might be one thing. Laws prohibiting courts from "kicking to the curb" non-custodial parents would make a big difference too.
So, I'd say first figure out if the non-paying father is really a "demon" at all. If he really is, then how to handle it is obvious. If he isn't, then maybe having a few people (including the legal system) try to figure out what they can do to help in get out of whatever money hole he's in (rather than demonizing him) would be a good start.