The Penguin is on to something. But will it work?
Does, elimination of private schools result in better outcomes for students currently in public schools?
I don't know how public schools are governed in the UK but where I live in the US, our public schools are governed at the municipal level. Local school boards, made up of elected citizens, approve policies, hire superintendents, propose school budgets, etc. for approval by citizens at election time.
Unfortunately, the composition of the school boards isn't very representative of the general population. By definition, the people who run for these offices are rarely working class. They are upper middle class and have the disposable income and time to perform these duties. Thus, the system continues to be biased toward upper class sentiments.
I'm afraid eliminating private schools would simply result in the elite parents invading the governance structure of the public schools molding things more to their liking.