I was in B.U.P.A. as a perk of my employment once. I seem to recall that all aspects of treatment are charged, not simply the consultant's fee. Drugs, ward accommodation, nurses' time, etc. My daughter had a tonsillectomy under the cover of B.U.P.A. and the real benefit was to be able to choose when the operation is performed which, in school time, is ideal. We chose half-term. If a patient's condition gets really critical they tend to be dumped back in the NHS system because there's all the hi-tech equipment on hand. Quite how that is charged I don't know, but if it isn't it ought to be!
Private schooling, it might be argued, relieves the state system of the extra pupils while the parents still pay for the LA schools through taxation.