Forget separating Church and State. Education in the form of almost all schools should be separated to the State.
A republic must be composed of adults that constantly challenge and question it's political leaders and demand justification for their actions.
Political leaders however, have a natural interest in seeing that the populations is docile, compliant and incurious.
Almost the best tool conceivable for the political elites of any stripe to achieve that goal is that they should control the education of the young.