I’m not sure why this comes as a surprise. Your figures show the “average” and, therefore, includes schools with very expensive things such as Olympic size pools, multiple sports courts, grounds to maintain etc. These are “nice to haves” rather than educational necessities. In addition, costs of running a private school are higher because they can’t reclaim input VAT, whereas state schools can.
Is there a reason why you haven’t pointed out the disparity across state schools?
“In 2021/22, education spending per pupil at schools in England was highest in Inner London, with approximately 8,860 British pounds spent per pupil in this area. By contrast, pupils in the East of England had an expenditure per head of 6,049 pounds, which was the lowest in this academic year.”
https://www.statista.com/statistics/381745/education-expenditure-per-pupil-england-region-uk/#statisticContainer
Rather than concentrating on a sector which educates 6% of children, perhaps we should be questioning why there is such disparity in the sector which educates the other 94%. Perhaps this disparity is part of the reason why some parents choose the private sector; because not everyone lives in an area with well funded, highly performing, state schools.