A lot of it boils down to people saying "We're paying because we want the best." (Some main subsets of which are: my child is too bright/sensitive/dyslexic/black/vegetarian to thrive in The Awful Comprehensive.) But most people (aside from a minority who don't value education) want "the best". Nobody ever says "well, I want a mediocre education for my child because he's a rather unimaginative plodder and a bit of a thug."
I know this isn't what people mean, but they must realise it comes over this way.
They do therefore realise that state schools are not packed with thugs and unimaginative plodders. That they do, in fact, contain Children Just Like Yours. And if it wasn't for the private schools, they'd contain even more Children Just Like Yours.
Sorry to spell this out, but it does seem to come as something of a revelation to some people.
I don't know how these misconceptions will ever be overcome. Maybe every child should have to do a term in a school outside their "chosen" sector. As a sort of exchange programme. It would be a bit like the French exchange, with the same set-up: stay with a host family, try out some of the local food and the local lingo, experience several moments of hideous social embarrassment, cop off with someone you'll never see again, and come away with a better understanding of the Funny Foreigners.
(I'm not entirely serious, before anybody starts blowing a gasket...)