While I think quality of state education is an important issue, it isn't actually the straightforward answer to why ex-public school boys run the country. The straightforward answer is that ex-public school boys dominate high level public sector positions of influence because the Government has chosen to employ them.
At every point in the state school system there are going to be children who are let down, but that does not change the fact that some children do get through the state school system.
The former Government was constantly on about quotas and getting more children from state school into university. Now, no doubt there aren't enough children from state school at the best universities, but it is still the case that every university in the country has more state school students than independent school students.
Even on the Oxford PPE course, most (but only just) students are from state schools. Cambridge has decided that these students are competent to do this course and accepted them. Even with the huge wastage of state school talent before university, we still have enough state school pupils entering top universities to fill every high level public sector job in the country ten times over.
They don't have those jobs though. The senior civil service is dominated by people from public school, as are many other public sector institutions. 85% of High court judges are from independent schools. Cambridge, Oxford and the rest are educating state schools students but they are not getting a fair share of the jobs.
So why do ex-public school boys dominate? Because our elected government keeps giving them the jobs with power and influence.
They don't have to. The people we elect could put quotas on government employees to ensure that state school pupils have 90% of the top jobs. It is utter hypocrisy for Governments keep blaming the education system and universities and yet keep on employing ex-public school boys when enough state educated people are already getting through the system and could take those influential jobs.