I haven't read each post so this might have already been said, but keep in mind, public schools are mandated by federal and state laws to provide services that private schools are not.
A child that is on a ventilator and in a wheel chair and needs a full time, one-on-one nurse, in addition to physical therapy & occupational therapy, the school pays for all of that (including salary & benefits for the employees, and any equipment that is necessary, even summer services if deemed necessary). A child that has a severe emotional disability and puts other kids or themselves at risk and needs to be served at a school out of county with the resources to handle such children - public schools can't just turn them away, so they have to pay their tuition and the special transportation to get there. A deaf child that needs a full-time sign language interpreter, speech therapy, teachers of the deaf - public schools have to provide all that.... (According to the Individuals with Disabilities Act, aka IDEA, and the requirement that schools provide FAPE, a Free Appropriate Public Education).
I'm not sure what the number is, but if you say you read that St. Mary's spends $15,000 per child, keep in mind that's an average. There are several children in the district whose needs may add up to over $100,000 per year so the others are getting "what's left" after the schools meet these expensive federal/state mandates under IDEA. When Congress passed the Act, they promised to pay 40% of the costs that providing these services would entail, but they have never come close to that (never above 17.7%, often much less than that).
So keep in mind private schools do not need nearly the staff the public schools require to meet all the federal mandates (special education teachers, physical therapy, occupational therapy, teachers for the visually impaired and Deaf/hard of hearing, adapted physical education, speech therapists, one-on-one aids, registered nurses, audiologists, teachers for English language learners etc.) so that might account for some of why so much of the budget is spent on personnel and why public schools are forced, by law, to pay for such expensive services these days..... they really do impact a budget greatly. And, cutting these expenditures is not an option, as the liability for not providing such services and the litigation expenditures in trying to justify it would likely be even worse...