Not That Kind of Libertarian: Puzzles of Children's Rights

McMegan writes:

I'm sorry if my nom de blog fooled you, but I'm not that sort of libertarian. Children are a perennial problem for libertarians, but what it boils down to is this: children (and to my mind, the severely disabled), have positive rights. They have a right to be fed, educated, clothed, sheltered, and given medical care on someone else's dime. And if their parents abdicate this responsibility, then it passes onto the community, including the state, even if none of us asked said parent to reproduce. So arguing that educating poor children is immoral . . . well, I hardly know what to say, except remind me not to get into a lifeboat with you.

I'm not that sort of libertarian either. But this is a problem. Children don't sprout from cabbage patches, as you may be aware. Here is the sequence of events, glossed over in Megan's argument, that must occur before a child becomes vested with a right to the property of other people.
(1) Coitus.
(2) Conception.
(3) The decision to carry the proto-child to term.
(4) Not putting the child up for adoption.
(5a) Abdication of parental responsibility.
(5b) Inability (for whatever reason) to meet parental responsibility.
(6) General positive right to be fed, educated, clothed, sheltered, and given medical care.
Does Megan propose the state confiscate children whose parents violate their special relationship-relative positive rights? If not, and such kids are simply subsidized, then doesn't this create an incentive for parents to violate their childrens rights so as to transfer the responsibility and cost to the state? If not sufficiently feeding, educating, etc. is a rights violation, shouldn't bad parents be fined, jailed or otherwise punished? If bad parenting imposes a cost on taxpayers, shouldn't bad parenting itself be taxed. But bad parents often have a negative tax burden anyway. What to do?
As an old Contemporary Moral Issues 101 fave puts it, if you have to obtain a license to drive legally, shouldn't you have to obtain a license to legally retain custody of a child after birth? What to do with the kids of parents unable to meet the requirements for a license. Shall we scatter them among wealthier homes like so much spice? Good idea! With an overburdened pension system, or scarcity of good labor generally, kids are a positive externality–but only if those kids are net taxpayers, not net tax consumers. So let's sterilize poor people, just to be careful, place kids of non-poor negligent parents in government programs/homes that will cultivate their potential for high levels of economic production (so our entitlement programs remain demographically stable), and give huge lump sum payouts for each child of couples with IQs two standard deviations above the mean.
I don't suppose that's Megan's kind of libertarianism either. Sam's Club Republicanism, maybe.
But seriously… having a kid and not taking care of it automatically entitles the kid to be raised by the taxpayers? Hmm. What happened to the intermediary institutions of civil society? Do we skip them? The state COULD give parents vouchers for food, etc. In fact, it does! (But not schools… God no! Not for schools!) But if these are in fact positive rights violating parents, do we really want to give them the vouchers? What kind of libertarian are you? I want answers, Megan. Answers!