I have seen the figures for many private high schools as well as colleges, and, yes, even full freight often does not cover the cost of having such a student at the school. Having a full pay student means less of a dip into endowment and other funds, and a greater pro oof getting some more money from the student and family through donations.
With private college cost getting up to the amounts where they are now, it’s going to be increasingly a wealthy families’ market, IMO.