I will speculate that there are marked differences between small, relatively inexpensive private schools (say, <$10,000 annual tuition), and larger, more expensive private schools (many prep schools, name schools with large, elaborate facilities). With the fancy buildings and grounds come a hefty cost in capital and upkeep, which requires them to recruit families from a segment of the population which is enriched for the elitism, entitlement, and toxic attitudes referenced by ultra. Sweeping generalization, of course.
Small, budget private schools, OTOH, are often eking along on fundraisers and multiple-hat-wearing staff and volunteers, which has the added benefit that faculty and staff tend to know the students from multiple perspectives. The schools also may be more likely to have some kind of common vision underlying their existence, whether it's religious or other-philosophical in origin. A school that prices itself lower is probably more concerned with access than exclusivity.
And, of course, all private schools select for families that have made conscious choices about their children's education, while public schools must take all comers, and cannot de-select challenging or labor-intensive students of any description
