In many parts of the US, the most accessible (i.e. not as expensive) private schools are Catholic schools, dating back to a boom of Catholic immigrants who felt (and were sometimes right) that existing schools (public and private) were anti-Catholic. There are non-religious private schools in the US, but Catholic schools are everywhere.

Modern US Catholicism has, for a range of reasons, become very politically conservative (particularly in comparison to the Catholicism elsewhere in the Americas), and because Catholic schools are generally attended by the children of practicing Catholic parents whose conservative views shape their view of schooling, these private schools reflect those political views. As a result, if you aren't Catholic, many Catholic private schools are going to look like political and religious indoctrination.