I cannot imagine that homeschool or religious-based schools will provide enough exposure to external information to produce more independent thinkers. I am not bashing either form. I am just responding to your concern about "hive mind," and I am using commonly understood characteristics of the three modes mentioned.
Then you have a very limited imagination. Schools do a poor job in general of producing independent thinkers because they are generally filled with people who are not independent thinkers who came from education departments that largely are rotten in the same way.
That said, even if they are in the minority, lots of public schools are pretty good. Lots of private schools are pretty good. Lots of homeschooling is pretty good. But to the extent they are good, they are good b/c they either have people who were not ruined by college, or that came with different experiences.
IN generaly though, the way we have schools set up is ridiculous. It sort of made sense at one time, but it's ridiculous now. For what we spend per pupil at even the worst funded schools, we could pool that money for 15 people, and pay one good teacher to teach them in groups of 5 for 2 hours a day, and we'd probably blow away the performance of most public schools. That would leave you without the daycare aspect of school but you'd get much better instruction.
Just for perspective, even in the worst funded school districts in Mississippi, that would leave you $105k to pay for compensation and benefits for teaching 15 kids. For Oktibeha COunty, they would have $195k. In the worst school districts, with the poorest people, you still might have trouble finding an adequate space to teach 5 kids at a time. But most places that would not be an issue.