Homeschool vs Public School
Public school as we know it in the West today, was originally designed to produce 1 type of person: cogs for a system that would require subservient followers. The format of public education is that of finding an answer from an authoritative source and it asks that parents trust that it comes from a place of caring for the individual's well-being. Public school produces a type of person that does not wish to think for themselves, does not wish to be independently-minded, and does not have the sense of curiosity and wonder associated with being able to distinguish truth from fiction.
Homeschooling is more of a philosophical belief in decentralization and self-education by families. There are organizations that offer, "Un-schooling" which essentially is just a raw search for knowledge, enabling but not restricting a student's curiosities. Anything under the sun goes, it is up to the individual parents to determine what their children will learn.
Homeschooling at it's core is the idea that no one should be given the power and privilege of educating the youth other than someone of their parents’ choosing. This should not be allowed to anyone besides those that have the highest probability of being motivated by the well-being of the student.
There is little evidence one could provide to change the views of a parent once they have decide between public or private schooling. If you believe that people are innately good and that people in general have each other's best interest in mind, there is little one can do to argue against public school. Homeschooling appears to some as a risk, as a dangerous experiment.
Public school, just like any ideologically motivated, legislated, mandatory activity, is a breach of the personal sovereignty of the family. On the other hand the law enforcing that all children be placed into government-run, ideologically motivated educational institutions can be viewed as a prison. If a parent does not put their child in a public school, the law has the, "right" to take action.
The argument between public and private in regards to education is an illusion. Free choice is the issue at hand. Are parents free to decide what is best for their child? No says the government. Why should the government defend their, "right" to educate and decide upon the curriculum that shapes your children's cognition? What could motivate such a policy?
The only motivation to restrict the free-choice of individuals is control. The government wants to control the education that forms the people of the future. Do you feel that the government is more likely to have the best interest of children in mind, or do you believe that parents do? The distinction between someone who wishes your children be forced by law to attend government funded and run schools boils down to a belief in the values of family or of control over one's self and children.
Understanding the difference and reframing the question is paramount. If you support the freedom of parents to educate their children then you must stand for homeschooling. If you believe that children should be publicly controlled because parents are unable to provide effective options and guidance then you must be in favor of public school. Make no mistake however that mandatory public education is a breach of parental rights.