1)Private education is the only answer compatible with First Amendment human rights and freedom of consciences.
2) All education is religious indoctrination. A religiously neutral education is impossible.
3) Parents, and the teachers ( tutors) they selected, would mutually agree upon the religious worldview taught. They **MUST** choose between either a godless or God-centered religious worldview and neither of these two religious worldview is religiously neutral in content or consequences.
4) As for wards of the state: It would be a very very very rare child that has absolutely no family. Even if it is the state who is paying for the care of the child, the nearest responsible adult in that child family would choose the religious setting for that child. ( Even if it were a 4th cousin five times removed.) The Mormon genealogy records could help locate these people. There is now DNA genealogy tracing, as well.