Schools can't turn a sow's ear into a silk purse but I do think, considering the constraints applied to the current GCSE/A level examining system, they can turn average students into A grade students. In fact I KNOW they can because several of my DCs contemporaries who did noteven pass the test(let alone gain a place) to the local highly selective grammar schools did suddenly end up with fantastic GCSE results, and I doubt they would have achieved that at a bog standard comp (NOT, I hasten to add at our local outstanding comp).
The point I am making is that 'average' students do get better results from independent schools, which is presumably what one is paying the money for.
Doesn't make them any more 'intelligent' though, just gives them better results, which isn't fair on those who can't pay.
I completely get the OP's statement. In our area, only the ones who don't get into the grammar schools tend to opt for the private sector and then they miracuously get places at 'top' universities....job done.
However, I do accept that this is particular to my area and there are many people who choose to send their children to independent schools for many and varying reasons, including those who wouldn't ever dream of using the state system,it wouldn't even enter their minds (eg generations who have always gone to Eton etc).