The best day private schools in racially mixed areas are full of the chidren of hard working immigrants. When she was 11 my daughter was one of only 2 girls in her class at Haberdashers who had 4 English born grandparents - no one else did they were from all over Iraq, Poland, Israel, India, Pakistan, hong Kong, Nigeria Korea etc etc If entrance is on academic merit then you do get a nice mix of different types of people and from backgrounds which is good.
However I've never understood those people who send children to state schoolls and say they know then a "real" world. In some w2ays they don't know a real world. They konw a world in some cases where peers can hardly speak and no one has ambition, a world where you might be expected to get pregnant at 15 or become a hairdresser (in some by no means all staet schools). if that is a "real world" the more a child is shown they can do better than that the better. Also all chldren have other activities and certanily no necessarily ponies and skiing. If you; in re a state school firs tyou will have children in your class who are not that wlel off, whole family contributes to pay the fees etc etc. Secondly you aren't locked into the school., you have friends out of school, you watch television, you go out, you see all kinds of people.
Indeed the privat schools arguably are better not worse at this. YOu will beiin the CCF etc and if not will be doing voluntary work so you'll be seeing OAPs and all sorts of people as part of your volunteering activities.
The ideal would be you emerge at 18 able to talk to anyone with confidence and interest.
(Just to correct the above certainly with the oldest children I wasn't a single parent. One was at or about to go to university when our long marriage broke up; and with the third for 7 years we opad 15% of fees as his father taught in the school and then I think had discount on the next school as he had a music scholarship but even I agree it was wise of me in my teens to consider the pay of various careers because it was that decision which meant the 5 went to good private schools and we moved too from the NE to SE and that also has a massive difference on people, their chances, exam results and all kinds of things - even the weather is better although it's pretty nice all over today)