Probably not his choice but that of his parents.
I don't see that his school background is relevant or a cause for celebration. He achieved his position as 'Chief of the Air Staff' (rather than the patronising & frankly rather insulting 'top crab') through his ability, had he happened to have reached the top job via an independent school on the back of the same said ability I don't think would be justification to condemn him - or though no doubt you do.
There are good and bad state schools but would you condemn someone who happened to go to the former? Is it only admirable to achieve success on the basis of maximum hardship & deprivation?
A good school, whether state or independent, can only do so much with the material presented, the individual needs the innate ability to have a chance of success. It's not simply a matter of purchasing a path to the top.
As many of the previous posters have said a great many parents are looking for a stable education for their offspring which would be impossible via the state system (although there are a few state boarding schools) rather than just inserting their progeny into the first rung of the old boy network.