I know what you mean, OP. My entire family were privately educated, my father is an academic, etc etc.
dds are at our lovely local state primary, for following reasons: we went to look at two local indies, and two local state primaries, and this school won our hearts for healthy sense of community, pastoral care, kids who were hugely enthusiastic about learning, and a good variety of families from all different ethnic, social, economic backgrounds.
I remember refusing to go to Oxford (where I grew up and was privately educated) because I felt I had only ever lived in a bubble...I don't want my dds to feel they live in a bubble, and the area of London we live in is prone to bubble-ish-ness, iyswim.
And yet, I get wibbly moments 
I would also get wibbly moments if the dds were in one of the local indies, though, and that I am quite sure about. Perhaps I'm just prone to wibbles.
Not sure yet about secondary