Okay, I am in an awesome position for this debate because I have experience of both private schools and state schools. For context: I am currently in a 'selective' private school (you pay fees, but you have to pass an exam to get in and if you're clever enough they'll give you whatever bursaries you need to go there). My younger brother goes to a very good state school.
Education should not be entirely government controlled, because the government has proven that they don't have our best interests at heart. Gove is a wittering imbecile and I am already scared about the influence he will have over my brother's and my education (he's already stopped me from taking January modules for A level next year, heaping the pressure on me in the summer, and came this close to forcing my brother into an eBacc programme that wouldn't suit him one bit).
You say that you don't want the wealth divide and therefore private schools should be banned. This is connoting that all private schools are only attended by the wealthy. OP, you are thinking of the type of 'private school' that only differs from state in that it is fee-paying instead of free. With that, I agree with you to an extent. However, most private schools are selective, filtering out the brightest kids and putting them in a competitive environment. THAT IS A GOOD THING.
I went to a state primary school and spent eight years of my life, in a classroom, bored to tears and learning exactly nothing. Some might say I was just lazy; my year 6 teacher genuinely told me, two months in, "there is nothing more we can teach her" - because they had been trained to teach to an average and I, like many other kids, wasn't the average. I am smart, (saying that is probably a TSR executable offence - oh well) but more than that, bad state schools treat you like you are just a number, a part of a quota, and that you are an inconvenience if you do not fit the average. I knew I wanted to go to X (my current school) since I was in year 3, because I was sick of not only being surrounded by people who didn't want to push me (and if people don't push me, I shut down and just sleep through lessons) and kids (and teachers!) who bullied me for being clever. That was not a bad state school, but it let down a lot of its pupils because the teachers were under immense pressure from the government to shepherd us through, get us through SATs, and cared about exactly nothing else.
At X, we are constantly in competition and being challenged to achieve things. It finally feels like I'm not a burden because I want to learn things, but that the school actually wants what's best for me. We need private schools because whilst we should never culture a rich elite of students, an intellectual elite absolutely should be cultured - on the basis of just how bright you are, and not your class, background, or personality. In all but the best state systems, the very brightest kids are left to their own devices in favour of those who can't handle it themselves, so the brightest achieve good things, but not their full potential. Private schools eliminates that problem, and can provide real opportunities to deprived, highly intelligent children who would never have had the chance to develop their skills.
My brother's school, on the other hand, is a brilliant state school. There is a huge, diverse mix of students from different backgrounds and cultures and abilities, and the school is streamed so every student of every ability gets the best care and teaching, without dragging each other down. The school has an ethos of getting everyone to achieve the best they can, but that achievement is subjective to the person. If all state schools were like my brother's, I would be very happy.
The wealth divide will not be solved by cutting out state schools. There is a HUGE wealth divide in state schools - the kids who can't afford designer gear, who are on free school meals, get victimised. Plus, and this is what you seem to be missing a little - NOT EVERYONE IN PRIVATE SCHOOLS IS RICH. NOT EVEN CLOSE. In my year, a minority of people are seriously rich. The average parent of a child in my year is likely in the same financial situation as the average parent of a child in the nearest state school. Plus, as long as intellect is the focus, bursaries are available. There are girls and boys in my year who are here on full bursaries because they're bright enough to be here but wouldn't have been able to afford it otherwise. I'm on a half bursary myself. The fact that I've had to work really hard to get into X means I value my opportunity more, and I value education more. There are many of my brother's classmates (even at his very good school) who take their education for granted and don't work at all.
In conclusion: we need private schools. We just need to have it so they are selective purely on the basis of academic merit, and they are open to everyone bright and hardworking enough to get in. And state school should not be seen as 'where the kids go who weren't smart enough to get into private' - the two sectors should be entirely on the same level in terms of teaching and facilities.