I don’t know a huge amount about school funding, but I believe that the figure of over £5,000 is the local authority element.
From the government’s own website and statistics:
“On a per-pupil basis the total funding allocated to schools for 5-16 year old pupils, in cash terms, in 2024-25 was £7,690”. Primary school funding will lower, secondary will be higher. That’s 54% higher than the figure you quote.
You are incorrect in saying that even the cheapest private is £15,000 a year. From schoolguide.co.uk
“the majority of pupils attend day schools with an average fee of £5,218 per term which equates to £15,655 per year”. That’s the average. Plenty are below that amount. The average is dragged up by the very expensive school day fees.
Whilst that is still obviously higher than the state funding, your figures are misleading at best and exaggerate the disparity.
Private schools also have costs which state schools do not. For example, they cannot currently reclaim input VAT.
I think most private school parents would be happy to pay the VAT on fees if they got a credit for the £7,690 per year per child that they are saving the state by not using the state education system.