Hmm, mirry, you see I doubt that most people send their children private for 'the teaching style' unless that is shorthand for small classes and selective intake.
I'm thinking back on conversations I've had with friends who went private. There hasn't been that many because I don't ask them to justify it. The one that I found best was a friend who said, very simply, "I was educated privately and so was my dh. We understand the system and are comfortable with it. I can't get my head around class sizes of 30. We can afford to give our children the best - or our understanding of the best - and so we're going to".
The one I found most irritating was one who talked about how her own (extremely posh - one of the very famous ones) school was packed full of teachers who 'had a real passion for helping children learn, and couldn't teach that way in the state sector which believes it's elitist to help children flourish'. Like she knew ANYTHING about state education, or about the teachers within it, or about how much more skill it takes to get 30 disadvantaged children to a place of excitement and enthusiasm, than it does to get 15 privileged children through any number of exams.
But most irritating is parents who refuse to acknowledge the role played by private education in social inequality. Who will not accept that they, or their dc, did not get to where they are today through talent and hard work alone. Again: why? My dd goes to a really great state primary, but frankly it would have to work at it to get bad SATS scores with the intake it has, and it does a great job of transmitting social privilege. Not as great as the local prep schools, but it plays its part.