I mean, schooling children at all is ideological: it's literally built on the ideology that we should have an educated populace. I'm really confused by this statement.
I think I'm missing your point here because you seem to not mind the indoctrination that is the basis of having a public education system in the first place, but you started with protesting public education's indoctrination in general, and then when you narrowed it, of course I agree with you because it's never worked that way because it'd be logistically impossible and therefore it most certainly isn't happening right now and therefore unrelated to your original protesting of indoctrination.