I don't think it works like that, really. I don't think most kids go around with a mental checklist of which of their peers were privately educated or not.
In fact, as stated above, there are a huge number of factors which determine how well a child achieves in their exams, whether or not the parents have paid is not the most important swing factor. There are world-beating comprehensives and appalling private schools, and many many other factors which come into play which are not about school at all, such as the child's intrinsic academic ability, ability to perform under exam conditions, mental stability, the amount of parental support received, peer support, tutoring, I could go on and on and on ad nauseam.
But also as numerous PPs have said, achieving good exam results is often not the key motivating factor for parents in paying for private school.
I think this thread is a bit of an attempted gotcha aimed at people who pay for their kids education, but you're grossly oversimplifying the factors involved.
Full disclosure I do pay for private education (not for academic reasons). I don't mind people having a pop at me for this for ideological reasons because I am uncomfortable with some aspects of the premise of paying for education, but ultimately my child's wellbeing is more important than commitment to an ideological standard set by others. If that makes me a hypocrite, so be it.
But also I think your logic is skewed and you're not really comparing like with like.