Abolishing private schools would simply increase the size of education budget with no advantage. What you need to do is to ensure that universities do not differentiate in favour of private schools - but probably in favour of state ones. This would deal with the 'old boy network' just as effectively as abolishing the private sector without increasing the demands on the budget. Remember that folks who send their kids to private schools are paying twice for the education - once through their taxes for the place they did not take up and a second time for the place they are using in the private sector.
Also, there is a need for boarding schools for kids of people who are likely to be posted abroad so that they can get the 'right' qualifications for English uni's and have continuity of education - and there is (so far as I know) only one state school with a boarding house... - at least we were told when our kids were there that it was the only one!