I know TJ got banned for his intervention here, but he raised a point which at least made me go and look for info - the allegation of an state subsidy of private schools.
It looks like City of Edinburgh spends just under £5500 per child per year. 25% of the city's population goes to private school. If the state had to school that 25%, it would add c. £100M to the city's education budget requirements. Finding a couple of sources which suggest £100M is the nationwide benefit of private schools charitable status leads me to conclude the state gets a good deal from private schools as they save the state more than they cost.
Private fees here seem to be about twice the state spend per kid, so not really a surpise there's a gulf in the facilities.
I don't know that it is the state's / school's role to do anything more than show kids what opportunities there are. Too many sports to cover properly so just getting kids active and pointing them at clubs seems to me the best option for the state.