It may be a case of him flexing his left wing credentials for the Labour Deputy leadership race, but Alan Johnson raises some good points when he argues that private schools should do more to justify their charitable status.
In some parts of the country private schools sits close to state schools who have limited games and PE facilities, yet in many case the state school are never allowed access to the private school's fields to use. This would in the first case be a simple way private schools could provide a service, but they could go further. Alan Johnson also suggest secondments, more twinning of schools and sharing staff expertise.
I'm not having a pop at private schools, but they see, to have inherited the title of "charity" whilst never seeming to have earned it. In the same way as hereditary peerages have gone, so should an hereditary right to charitable status, unless it has been earned.