I'm not referring to studies arguing for abolishing public education. I'm referring to studies that show homeschooled and unschooled kids can get an education without school. Formal school is one path that works more or less most of the time. It's not the only path. Homeschooled and even unschooled kids do manage to get into universities and graduate at a rate comparable to or higher than public schooled kids.

The reason "homeschooling can't work for everyone" is mainly economic and social, not because public schools have a magic technique for teaching. Many parents need schools for day care so they can work, and many parents don't have the patience, time, or interest to help their kids educate themselves. Public schools have the same problem at a much larger scale: plenty of kids drop out or graduate with poor educations, for a variety of social and economic reasons. As I mentioned already the number one reason teachers give for poor student performance is lack of parental involvement. It should be obvious that parents who choose to directly take charge of their child's education have shown their commitment to parental involvement in that process.

For most of human history formal education didn't exist, and literacy was not a requirement for functioning in society. Children learned necessary skills from their parents and the wider community. If they needed to learn carpentry they apprenticed with a carpenter. Elites could afford to educate their children, but those educations were more for maintaining social status (dancing, fencing, manners, music, poetry, etc.). Education in the modern sense was not available nor particularly valued or needed until, well, modern times -- post-industrial revolution. Compulsory education is even more recent, though Plato advocated compulsory education for social control (as opposed to valuing education for its own sake).

If you're referring to the aristocracy sending their children to monasteries or tutors, you should look into that history. One of the main reasons some children ended up studying at monasteries or nunneries was primogeniture, in effect getting "extra" children out of the way without making them learn a trade. Aristocratic children were schooled in necessary skills (skills we today would not accept as a well-rounded education) not because they would ever have to get a job, but so they could make a good marriage with another aristocrat. Pretty much by definition aristocrats don't have to work so education was a luxury and social currency.

Homeschooling doesn't mean the parents do everything. Homeschoolers typically engage tutors, mentors, classes, etc. I couldn't teach my kids everything, they had tutors for things like music, other adults for subjects I don't know enough about, classes at colleges as they got older. It's possible to pick and choose, lots of ways to learn that don't fit into the mold of formal schooling. Most adults will admit when pressed that they learned their most valuable life and job skills outside of formal school. You only need about 50 hours to teach a child to read, write, and do basic arithmetic. Once they can read they can direct their own learning with some guidance. Kids learn very naturally on their own. School is one environment for accomplishing that but it has a lot of baggage and not a great success rate (except in terms of providing taxpayer-subsidized day care and lessons in conformity). Other learning environments also work.

Unschooling at its core means teaching children how to learn on their own, a skill more valuable than any subject in a formal curriculum. And that's a skill schools actively discourage, because it would interfere with the whole program.