An Argument for Intervention
I have a unique perspective on things, given my social and political philosophy background. I think a case can be made for intervention even in cases where there is no suspected abuse or neglect (I won't argue for those, since I take the reasons intervention is necessary there to be quite clear, although the same arguments I'm going to apply here will also apply to extreme cases).
The purpose of societies will differ, but societies as a whole will suffer should their populations suffer. Without a good education, there are not solid foundations that give an individual the ability to participate in the society. No matter what economic structure a society has, there needs to be a means of production. Societies need people who are well-trained in their professions and trades. Prior to the public school system, there were tutors and apprenticeships.
If a person lacks the resources to participate in society and the economy, that person will then need assistance to do so. When this occurs—whether the person is relying on family members, a social safety net, or is scraping by through illegal means—that person then becomes a burden of some sort or another and must now somehow be dealt with by the social justice system that has been set up by the society.
If a high number of adults are unprepared to participate in society, the society may crumble. Because of the burden created on society by individuals unprepared to participate in society, as with the case of threats such as violent crime, society is then justified in an intervention.
Parents who lack the basic resources, education or motivation to effectively educate and prepare children to become adults able to participate in a society and its economic structure are thus causing the society to struggle.
There are some homeschooling parents who are not capable of properly educating or preparing their children for adulthood. These parents may suffer from debilitating physical or mental illnesses, may lack education, or may not have enough of a social network to perform well at the function of educating their children.
Therefore, there are some cases, where a society is justified in intervening in a parent's right to choose the best educational method for his or her child. By the same premises, one can also justify the society's interest in education and right to intervene in any school that is not performing its primary purpose - to prepare a generation for participation in a society and its economic structure.