All the emphasis in this thread seems to be on the financial aspects. I know people who have borrowed heavily to put their kids in private school, and I know multi-millionaires who have their kids in state schools, so its not all about people do it if they can afford to and don't if they can't. The two types of education are different, and there are many criteria that people use to determine which is best for their child.
Cnofesd mentions that some parents have a keen interest in education and spend time with their kids, and some don't. This is true in all income sectors, and in all geographical locations. Even a mum who works can find an hour to spend reading with her child if she really wants to. This is where the "inequality" lies, not just in the schooling. 90%+ of a child's attitude to learning comes, in my opinion, from their environment, not just at school but at home too. Parents who have had a great experience with education themselves will undoubtedly pass on their love of learning to their children. Perhaps if those parents had a good education they are more likely to have a good job, more money....perhaps they will then be in a position to choose to privately educate their children if they feel that's best for them. To outlaw private schooling is not going to stop mum's like Cnofesd's mum sitting and teaching her child to read, and why the hell should she stop, so the woman next door who can't be arsed can feel better about 'equality of education'? Come now.
So to remove choices regarding education does nothing to address the range of abilities, opportunities, or love of learning that there is in our society, there will still be people who do better at school than others (and some of the factors which decide that, whether you like it or not, will be financial).
Instead perhaps the state sector could look at what it is about private education that's good, and see which bits of that they can model. I don't think that the state sector has it right and that private schools are better, I think the private schools have it about right and the state schools are lacking.
And let's not get carried away with the idea that private schools have loads of money to play with, many don't. If they are 100% self financing which many are, they can pay over 80% of their income to pay their staff, then there are buildings to maintain (old buildings in many cases) and all other expenses to be met, some quite large schools have a huge turnover and very little profit. Their accounts should all be available to be examined if you don't believe me. If you want to see state of the art facilities, a new sports hall and a whole suite of new computer equipment, you are more likely to find them in a state school than a private one, that is definitely the case where I live.