Private schools are private. They usually do not receive tax dollars or state/federal funding. Their parents still have to pay full taxes to support public schools while paying tuition to the private school for their children's education. They get no tax break. 

The point: Private school is a choice. No family choosing private school is forced to choose it. They choose it, at great financial sacrifice, based on their personal values and the rules and values of the school itself. If a private school chooses to mandate vaccines, that is fine. Parents can choose to change schools. Likewise, if a private school opts not to mandate vaccines for staff/students, that should also be acceptable. Families can choose to withdraw their students at any time if they are not in agreement with school medical policies. The families and staff know the risks, choose the risks, and are free take whatever personal measures they deem appropriate.

* Note: I am part of a private school. We wear masks, we clean, we distance, we quarantine individuals and their contacts according to Virginia's Health Department guidelines.  We changed procedures and methods to be safer. In well over a year of in-person classes, we have had no big outbreaks, no serious COVID issues. Perhaps this isn't possible in large public schools, but it is possible in less-crowded private schools. Our record is evidence: we can succeed without a mandate. Our students did not experience the learning loss or mental health issues that so many public school students experienced. This is true of many private schools. Our staff and our students should be exempted from a vaccine mandate.