The connotation of a private school is very much based around its agenda. In the 90's, the parochial schools (at least in NNY) really started pushing high educational values and not so much of the Bible thumping (although Roman Catholic theology was an important subject), and people flocked there. Once they started Bible thumping after 2000 or 2001, they started to contract, and eventually consolidated or folded. Sometimes there are other reputations, like how the parochial school near where FlagDUDETTE grew up has a "Please Sister, not the ruler!" reputation, but I don't know anything beyond that.
A Catholic school bible thumping? Surely you jest!
This is from somebody who had 11/12 years of Catholic education starting in 1961.