I'm not totally certain what my views are yet, but maybe private schools are better for some types of children. If your child is particularly academically minded and could cope with a higher level of work than the average level in a state school class, and is bored at state school, then maybe they would be better suited to a private school.
If your child is really good at something and that is the thing they will want to progress in throughout their lives, should you feel you have to hold them back by leaving them in a state school where the teaching has to be at the average level of the abilities in the class? Should you feel you have to do this just to make sure you aren't being a snob?
My children are of an academic standard of average or slightly above, but not high above average, so I don't feel it is a priority to give them opportunities to study at the 'top level' in a class of very academic pupils with no average or below pupils dragging the level down. For me the priority is for them to get a reasonable education and also benefit from learning more about society by mixing with people from lots of different backgrounds, richer, poorer, and different cultures. I don't think children at private schools learn so much of this side of things.
Also, if my kids could only get into a shit state school rather than a good state school, then I would worry about the level of teaching being dragged down too far below average for my average kids. I would also worry about them learning more bad behaviour from more badly behaved kids. I would then consider putting them in a private school.
I agree that the academic level of teaching and the exam results are probably higher in a private school, and probably the discipline is better if 'bad' kids are asked to leave as people have said, but the learning about people and society and real life is probably not as good. If there was a choice between a good state school and a private school I would choose the state school. If there was a choice between a bad state school and a private school, I'm not sure, but I would definitely consider the private school. If my children were super academic 'types' and that was what they were into then I would consider private school so that they could progress quickly in what they love.