The "school choice" and "religious freedom" angle has been at the heart of the modern American Christian Conservative movement since the jump (especially post Brown).
Interestingly enough- public education was a hallmark of the social gospel movement in the second half of the 19th century- Christians who saw their calling in educating kids who were otherwise working in factories or on farms, leading to public schooling (contemporary of Mann and Dewey) and child labor laws.
Of course, American fundamentalism (and evangelicalism) were responses to the social gospel movement... so the Religious Right's pushback and the modern equivalent certainly make sense.
Ironically, families who are likely to reject "gub'ment" being involved in education are often from economically depressed areas where other schooling may not be possible. Ah, well... better to be ignorant than "indoctrinated"