This is how a significant portion of the population gets radicalized by their parents. It needs to be shut down.
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With that attitude you might as well just tell parents that they shouldn't participate in society!
Education systems as we know them today are absolutely about indoctrination in so many ways. Capitalism, love of country, views on family units, beauty and aaesthetics, what has cultural value and what does not etc etc. Not to mention many school systems just straight up having classes on religion, allowing armed forces into schools to recruit and the like.
Whether you're worried about left wing or right wing indoctrination, it still holds true. All kids are being indoctrinated every time they go to school same as every time they watch TV.