TJames, you raise many important points, but your main conclusions are not valid. You are right about at least one though, you were unacceptably wronged by a parent. That injury has caused you to fall into the trap every demagogue dreams of occurring, you want to subject others to your fear.
Your experience with homeschooling was so bad, you are willing (even as a self identified libertarian) to defend the statist schools. Schools which make possible the indoctrination that allows American wars and have resulted in the deaths and mutilation of millions. The demagogues rely on this strategy for every statist ploy. Fear the terrorist, let the NSA do what it wants. Fear the diseases, force everyone to have vaccinations. Fear your own incompetence, let the state have your children. It is your fear and you want to SUBJECT us to it.
You do not explicitly endorse the use of force against the consent of others. You say that you would be willing pay to prevent inadequate homeschooling, not that you want me to have pay it. Except that you want “take over” the statist schools. So until the statist schools are no longer funded by tax dollars (the only valid way to do it), you do want to force me(or some other person) to pay for it, at least for little while.
The second biggest mistake, after giving into your fear, is assuming that parents don't own their children. Persons as property is inflamatory because of slavery, but it never really went away. Ownership implies to right to use. Whoever can tell a child what to do owns that child. Can you tell a child to eat their vegetables. Can a teacher in school order a child to stand up in front of class and read from text? Changing the words won't change the meaning. Someone owns children.
If a parent doesn't own their child, then someone else must. The only other real option is the state. In a free society power is delegated to government from the people. If the people can't own their children how can they delegate that power to the state (By the way this argument works almost any property rights issues: guns,which have I used successfully on an Obama supporter; drugs; nuclear weapons; etc...).
Ownership in this case is not absolute. Because the ownership of a child passes from the parent to the child at the age of majority (or during an emancipation or manumission proceeding), the ownership resembles a trust. The parent is the lawful owner (trustee) and the child is the equitable owner (beneficiary). The parent is required to make sure that when the ownership of a person passes from parent to self, that person has a fully functional self. As long as the person comes into their self with a reasonable amount of opportunities fulfilled (proper education, proper nutrition, enough wealth to start a life, etc...) the responsibilities of the trustee (parent) are met.
The proper role of the organized use of force (also called government) is not to tell people what to do, but to let people know what they can't do because those actions cause harm. It not my job, or governments job, to tell anyone what a child must do or learn. It is legitimate to identify what is harmful. A child coming into adulthood without work skills is certainly harmful. Whether a child is trained as cashier or a doctor is not really relevant. How to measure what is harmful to child and how to codify those harms in way that can be enforced in free society is daunting task (maybe age appropriate standardized testing). It's also not my responsibility. It is the responsibility of anyone who tries to use force to protect a child to get it right, or face prosecution for unreasonable search and seizure of property.
TJames, if this is your passion, we could use more libertarian social workers. (Teaching them to fish and all that!)