Thats not really true though unless you are just looking at Eton, Harrow, Winchester, Westminster or Fettes where its a minimum of 30k+ per year.
There are lots of other public schools (fee paying independent schools) which are less well known that are around 15k per year. These are more than affordable as my nextdoor neighbour, a chartered accountant, who sends her 3 kids to a nearby independent school, would attest.
If the schools offer a significant number of scholarships which allow gifted kids via entrance exams, regardless of financial means, a place to get a high quality education then maybe charitable status is valid. What a significant number of scholarships is, well thats open to debate, but to me it wouldnt be unreasonable to say 15%.
Those however that are only open to the exceptionally wealthy should be taxed fully and stripped of charitable status.