I think that, except in aeas where for some reason the state schools actually are objectively failing, parents who choose private for their capable, non special needs child end up choosing a different culture more than they do a 'better' education. Certainly at Primary level anyway.
I seriously doubt there is any difference in educational value between the private prep where I work and have my children and the outstanding village primary school over the road. But at the village primary the children arrive at 9, have their literacy and numeracy hours and other lessons as a whole class in their classroom, play in the playground at breaktimes and go home with their parents at 3.15, maybe to an after school activity or maybe to spend some time with their parents. Lovely. At our prep the children arrive at 8, travel around the school to different teachers for different subjects in different sets, do clubs/extra lessons or play in the woods/astros/golf course at breaks, have sports coaching after lessons evey afternoon, head into Prep after sports and to tea after that. Many go home at 6, many stay for after school activities or more Prep (for the older ones on certain days). Many stay on all night as boarders. Also Lovely. But very very different.
There are things about the primary day that I prefer and things about the private day that I prefer. Swings and roundabouts. I don't think you can say that one school is 'better' than the other, they just suit different children for different reasons.
However, I do think most private schools can provide a better education for the very lowest and te very highest ability children, purely because they can be taught individually and in tiny groups with more ease. Our brightest children are streamed into a scholarship form of 8-10 children at 11, preparing them to win scholarships to top public schools. By age 13 they are working to GCSE level at least in most subjects. Our weakest children have individual lessons in English and/or Maths once or twice a week depending on need and are rarely in a set of more than 10-12. Unfair though it undoubtedly is, these children are stretched or supported in a way the majority of state schools cannot manage.
It's also inaccurate to say that 'an A is an A' from all schools as actually some independent schools take iGCSEs and many take the IB which is harder (I think, unis seem to think so anyway). Though having siad that, I think state schools are starting to adopt the IB.