I want my kids to see a realistic representation of life in their school, not the bubble of privilege that their professional parents might well be able to create.
This is what worries me. I went to a private school (abroad, I'm forrin) in a deprived country, and the gap between the day to day lives of the pupils and those of ordinary people was unpalatable. A new car was a very "normal" 18th birthday present / bribe for decent final-year results, for example. I worry that the gap would be as wide in London. I don't want to raise an entitled wally (does anyone?).
Friends and acquaintances who have gone private seem to be working all hours and making sacrifices to pay the fees, and - barring my children having specific needs that are served by the private sector - I don't want DH and I to be caught in this trap.
Complicating things further... DH was raised in what we'd call a council estate in the Soviet Union, and went to a correspondingly grim school. He now works at a company whose products you use every day, alongside graduates of Eton, St. Pauls, Westminster (and subsequently Oxbridge) etc. I worked in City law alongside people from fairly diverse backgrounds, in spite of the reputation the City has.
In all honestly, as we're house hunting, I expect we'll go with a "selective by postcode" state school, that we like the ethos and feel of. Which feels like a good enough solution to me right now. Long term, no idea.
TL:DR - sympathies OP. I'm not sure I have the right answer either but I think your approach / assessment on the merits of particular schools for particular DC, is the correct one.