Well, if you don't know of a state school with an entirely middle class catchment you haven't seen Collis school in Teddington!
You need to be able to afford Teddington house prices, but not having that £1m spare I don't live there.
We send DD private and I know full well that there are parents of Collis kids who earn about 3 times what myself and DH do...but they must have huge mortgages. I've decided against a mortgage and pay school fees instead.
Interestingly in the state school infants DD was at prior to going private, in a school of 300 children there were 3 black children, 5 Asian Children and one Chinese. Her private school is nearly 50% ethnic minorities, and certainly is not a white middle class ghetto, despite me living in a West London white middle class ghetto.
But I suspect W London severely distorts the national average when it comes to schools, the catchment areas are about 1-2 streets near the school and that's it.
We knew that we could get DD into the local private school more easily than we could move house in order to have a chance of getting into the local state school. It also worked out about the same in money terms and frankly, I didn't want to move house.
I'd really prefer not to pay school fees but I don't think feel I am doing a terrible thing opting out of state and those people who lie about where they live or rent a house for 6 months just to get a school place are doing a lot worse, they are taking a place away from someone, I'm just deciding not even to bother applying for that place.
So , I think the Op is NBU.