This is such a good threat @Questionwithanamechange
I know many families who chose private I think the biggest difference is on the whole (and of course there are exceptions) they've enjoyed their education, developed wonderful relationships with school staff, been exceptionally motivated, had strong people/public speaking/social skills and had contacts that meant they walked into opportunities sooner and through less of their own doing, than state school peers. There will always be exceptions, but I think there has been much research to show there are differences though kids can do well anywhere in theory. And many do.
Ironically my niece went to a wonderful private school from age 7-16 and then left to local sixth form. While all her private school friends have gone onto wonderful gap years, and excellent places at Uni. They've done voluntary work and are just fabulous kids all round - very nice and down to earth too. Niece has hugely struggled, isn't interested in education at all, did the bare minimum, and pretty much works in a restaurant a few months a year then quits. She has wonderful parents who have probably spent 350k on her education. However, what I would say, is I think deep down niece has always seen she doesn't really have to apply herself. She is set to inherit a large house, a buy to let flat and a house in France. Her parents are the most down to earth people, but nothing they've done has ever made their daughter care much about her education or future. They come from nothing and worked bloody hard to get dd the opportunities she's had. But it hadn't changed her level of motivation and I think knowing so much was in her future she really couldn't see need to work. Having said that she had a wonderfully supportive education, has made wonderful friends and didn't get into some of the negative trapping that would have happened had she fallen in with the wrong crowd. Private schooling won't fix every situation. But even here I can see how she benefited.
If I was in your situation I'd probably try to do a buy to let flat, thinking I'd sell it and split between two dc for downpayments in 25 years. But not tell them. I'd also only do that if I could see they worked hard and were saving themselves - whether they are bus drivers, gardeners, Doctors, Teachers or anything else.
As for schooling I'd do private primary unless they got into exceptional local primary (they do exist, my other dn are at one that is truly a gem). And then attempt grammar schools for 11+ if they meet the needs of the child. A dear friend who is a Consultant, only 2/3 of his kids got into private secondary, the other 1 got into a great grammar school and he is very happy. He's quieter, less academic though still very smart and is thriving.
I think by doing younger private you open up more options for secondary whereas I know several people who did state primary and couldn't get kids into secondary private nor good grammar schools. I think by educating private at primary level you are starting education off on a better foot and often that means you have more options down the road. Plus I think little ones thrive in smaller class sizes and unless there are exceptional needs that won't be met in a private school, then have the best chance of loving education too.
Our local primary schools have had terrible incidents of bullying. And I agree are far more keeping up with the Jones. Many private schools have middle class parents sacrificing hugely to make it work. As well as many whose grandparents 'have to' help too.