"Separating ideology from education is impossible."
If that is true (and you haven't made the case for that yet), then you can choose the ideology, and make it explicit. Not all ideologies are equal.
"The ideology (or religion) ..."
Are you saying that the words "ideology" and "religion" are synonyms?
"...of public schools is whatever is in the state's best interest: flag worship, military recruitment (as required by No Child Left Behind), "school to jobs" (as if that is the main purpose of education), psychological profiling (see Cloning of the American Mind) and psychological drugging."
So you assert. Yes, it "can" be those things... just as private schools can have some ideological agenda. What's your point? You seem to be automatically assuming that "State interests" == "bad"
"I'm more of a Distributist than a Libertarian."
What is a "Distributist"?
"I do believe the state has a right to tax to help clothe, feed, and shelter the needy. It is quite easy for the state to keep these functions separate from religion. As I wrote above, it is impossible for the state to keep education separate from religion,..."
You wrote it, but you didn't make your case for the truth of that assertion.
"...and furthermore the rate of taxation is so high as to limit the choice of those who seek a religion other than the religion of the state."
How much is "too high"?
"As an archbishop of the Vatican stated last year:
"Even in states in which the right to religious freedom is taken very seriously [...] limits on religious freedom exist [...] For instance, government and taxation policies may limit the rights of parents to choose a religious education for their children [...]"
There are always limits, since we lack an unlimited supply of resources. The question is, where does a State/culture set those limits? By providing a public, religiously neutral (I don't accept your assertion wrt the equality of religion and ideology) education, we provide a common baseline for everyone, and which everyone benefits from, whether they have children of their own or not. If you want something above and beyond that, you pay for it.
"With distributism, non-corporatism, and a smaller government, monopolies (state or corporate) would not be dictating education."
What makes you think that? If McEduCorp owns all the schools in your area, and has the money to erect barriers to market entry, they damned well can dictate education... and without State regulation, there isn't a damned thing you can do about it unless you can match McEduCorp's resources.
"The "free market" (constrained intentionally to keep it free from monopolies of either type) would dictate what types of schools there would be."
What would keep the market "free"?
"Under such a system, if there are enough people who believe as you do, there will be secular schools. If there are only religious schools, then that just reflects the community standards. Even in that case, it would not constitute a theocracy, as it was the (monopoly-free) free market that dictated it, not the state."
So long as someone with more resources than I can control my access to resources, a market won't remain free for very long. So, unless you have the tech to create a Post-Scarcity economy, or you are willing to settle for a low population density and low level of resources, the person with the lion's share of the resources can leverage that into power and control over the rest. I can go into more detail about that if you're interested.
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