OP we're a few years ahead of you, with a bigger annual income/savings pot, and we had the same dilemma. Obviously our experience is personal to us, but I thought I'd chip in as we were in the same boat (me wanting private, DH saying state).
We went state, despite offers from some of the most best public schools, because our local state is excellent and because I couldn't see that the advantages of public school were worth what would be a considerable outlay to us. I appreciate if you've loads of money, it wouldn't be a sacrifice, but it would have been to us.
I am so glad we did what we did. We are financially comfortable, and DC are happy. We do organise a tutor in one subject, which costs £40 a week (central London) and DC gets themselves there on the bus, so all I have to do is put £40 into an envelope and put it in their school bag once a week. No drama.
DC have really nice, educated, well-mannered friends (we went to public schools, as did lots of the parents of children at the state school), from a very broad range of backgrounds, which I love. Some parents are lawyers, some work in high street shops, some don't work.
When I compare DS's writing and spelling and reading ability to his cousin of the same age who has gone to super smart private schools, our state-educated DS is way above (so satisfying).
I think private school guarantees you friends from a certain social level, and to those with new money that is probably a big lure.
The opportunity to do loads of things eg try the sax, is also going to be better at public schools.
But honestly I am so so glad we went down the state route, and thank god that we did.