Yes, I find it 101% unlikely that the people I described above would send their children to any state school even if it was truly brilliant. They know what extras their fees are getting (including, as it goes, close relationship with teachers - whoops 'masters' who then can write personally to the Oxford college their offspring apply to recommend their DC). It's a literal Private Members' Club.
This doesn't apply to the school DH works in to the same extent (once called the most expensive comp in England!) so there are shades - but aforementioned DS was pretty much sidelined in his cricket team by those boys who all went to the two local private schools, particularly the more expensive one.
People I speak to who send their kids private and my own experiences tell me that the thing expensive private education buys is a way of speaking , behaving and navigating the world - a kind of super confidence that state schools do not have time or curriculum room or class sizes or pastoral and extra curricular (or co curricular whatever that means) space to enable. And also Ofsted and league tables and shizzle.
Full disclosure - I went to a very highly regarded private school in Glasgow (on a bursary if it matters) . There were reasons for this tied up in moving house. I had brilliant teaching and opportunities. It really is a genuinely excellent school. And there were lots of different types of people there but vanishingly few people from really ordinary backgrounds. There was a definite divide in the school, between the people who started aged 12 ish from a state school and those who had been at the school from aged 5. The school is very very cash rich.
Even though I went to a fee paying school, I and my friend who went to a genuinely posh Edinburgh school had backs actually turned on us in the first week of university by some young women who went to a small set of south England based public schools.