Most private schools are still required to follow a state mandated curriculum in many areas, and are still hiring from the same pool of indoctrinated teachers.

Most private schools are still required to follow a state mandated curriculum in many areas, and are still hiring from the same pool of indoctrinated teachers. You could make a propaganda light private school, but that would need to be your explicit goal. Sure, children will encounter a diverse range of ideas. Schooling dominates in time spent with, and it is the one most likely to reward or punish a child for regurgitating an ideology.