Whilst I agree about tax loopholes, I have mixed feelings about private education (to give it it's proper designation). You see, I had the benefit of a similar education provided by a charitable institution that cost my mother, a war widow, nothing. The only way that institution is able to continue in existence is by providing education for a much larger number of fee paying pupils. It still provides free education for children from deprived backgrounds.
As for the preponderance of public school educated individuals in certain professions, whilst there is strong evidence of networking between the families involved, we cannot rule out the possibility that the nature of the education provided by such institutions is in fact more suited to those professions than that provided in state schools.