Should private schools be abolished?

It is an inevitable feature of democracies that the rich have particular access to politicians and policy-makers. While the rich don't have a need for public education because they can pursue education for their children from other sources, they have no motivation to lobby politicians on behalf of the education system and a perverse incentive to remove education from political agendas in favour of their preferred issues and legislation.

Only by forcing the rich into the same situation as the poor can we expect to gain meaningful ground in terms of education reform, especially in terms of increased funding relative to national and municipal budgets. We cannot expect education will be a national priority until the entire nation has a vested interest in the good order of the system.

By "forcing the rich into the same situation as the poor" there will only be resentment, and the most well off will send their children abroad to more democratic societies where freedom of choice is allowed.  
This proposition argument assumes that those who are less well off do not have the motivation to lobby Parliament, which is an insult to those who work hard trying to implement education changes, and send their children to state schools.