Great post, Luckygirl3.
I would definitely remove charitable status from private schools. The only reason I would stop short of banning them altogether is that doing so would leave the system entirely under government control.
I would like to see an overhaul of the system to bring in some of the things I mentioned upthread. I believe that all children should be taught to understand how politics and the media work, and that the mocking of these subjects (along with sociology) is deliberate, to discourage people from understanding how they are being manipulated. I recognise that it is important to remove bias from their teaching, however, and that checks and balances need to be in place.
I would raise the status of practical skills-based subjects such as plumbing, childcare and catering, perhaps by insisting on all university entrants having to have at least one A level equivalent in a subject from a list. Many courses insist on English and Maths, even when the course has nothing to do with either, and Latin used to be a pre-requisite too, so there is precedent. I don't think that just saying 'oh, practical subjects should have the same skills as academic ones - your children should be able to study BTECs in carpentry and hairdressing, and mine should go to university to study classics and medicine' is enough. We need to mean it and enforce it.
I would prefer all education to be free for those who will use it, but if it can't be provided by the state I think that vocational subjects should be paid for by the employers who profit from them. At one time that happened, up to a point, when large employers would send people on day release, night classes or sandwich courses, but the burden of fees has transferred to students, who now pay for the right to earn salaries making profits for others. I believe that the study of subjects such as literature or history are also beneficial to society, as they teach the sort of skills that can be applied more generally, and without them we would be a country of barbarians. It's less clear who should pay for them, though - making those students pay, whilst making employers pay for STEM courses would put those courses out of the reach of many, and I wouldn't want to see that happen. It makes more sense to me for everyone to pay for each other, but in that case there should be no elitism and denial of opportunity meaning that school leavers pay for the education of those who will always earn more than they do.