People are making a choice for their kids, whether they want the after effects of religious or public school 'brainwashing'. For many, its a clear cut choice, one way or the other.
And society should also care whether government-run public schools are doing this--not to mention whether they are indoctrinating children with values that many parents disagree with.
School boards are elected, but I don't think "broad democractic oversight" is a fair description.
In any case, none of the things you mention prevented the US public school system from being explicitly set up to indoctrinate children, as explicitly described by the people who set it up. So to the extent that society in the US cares about educating children, it cares about educating them according to a particular political agenda that many people do not agree with. That is a major reason why homeschooling continues to increase in popularity.