This was a loophole discovered by the homeschoolers - the state could ban homeschooling but the bar for registration as a private school was quite low.

My kids are in their thirties, so it's been a while, but my recollection is there were three legal means to homeschool and registering as a private school was one of them. Another was to hire a tutor for three hours a day.

I think the third was probably to go through an umbrella school, which my family did initially.

Homeschooling laws vary by state. Some are more stringent than California and others less.