"I think that MrsJAlfred's points are the clearest argument I've seen so far in favour of not shunning private schools, but actually undermining the system from within hasn't really happened so far, has it - unsurprisingly: why would you want to destroy from within a system which has probably served you rather well?"
Who said anyone wanted to destroy it? I don't want my sons to destroy their excellent school, nor would they want to. I sent them to that school because I want it to be possible for them to become MPs, dosctors, judges, leading journalists, FTSE CEOs etc. They completely understand they've received a significant advantage in going to the school they do. I hope when they're up at the top table influencing policy, they'll make sure that it isn't only the elites who are able to take up positions of power. I hope I've drilled into them a sense of responsibility. But I don't expect them to adopt my poorly arrived-at poltiical ideology and run with it, they'll have ideas of their own, meaningful to them.
And if they go to ratshit and become investment bankers at least they'll have their love of Latin and memories of singing in the Abbey to see them through their dark, lonely days. 
As anyone who's read Sutton Trust reports knows, attending a grammar school confers significant advantage also. Who in their right mind would say: "On behalf of my daughter I decline that significant advantage offered, and I'll accept for her that significant disadvantage over there." At least seeker wasn't bonkers enough to do that. But if you have already accepted advantage for yourself, generously provided courtesy of the state, is it really fair to deride others for securing similar advantage? Assuming it's paid for out of honestly-earned, taxed income?