'Stay out of religious schools' seems to mean just that //

This is how he demolishes the claim pushed by the Morrissey-Berru and Biel that they did not serve as ministers of any kind:

They were Catholic elementary school teachers, which meant that they were their students’ primary teachers of religion. The concept of a teacher of religion is loaded with religious significance. The term “rabbi” means teacher, and Jesus was frequently called rabbi.

And finally

…When a school with a religious mission entrusts a teacher with the responsibility of educating and forming students in the faith, judicial intervention into disputes between the school and the teacher threatens the school’s independence in a way that the First Amendment does not allow.