Huh?People over 66 and some others do not pay NI contributions but receive NHS treatment free of charge. Much of NHS funding comes from general taxation anyway.
Saying that the private school should pay is like saying that a private dentist, that you have a regular payment plan with, should pay a NHS dentist you happen to go to for some reason.
It’s not complicated. If the school has already been paid for the education of that child, then they should provide the service paid for. If the school has an issue with that, your beef is with the school.
It’s still not clear what your answer is to my question about people choosing, or not, to be part of the NHS.