The "British Establishment" is something I'm still not sure I understand. I also don't know whether it is distinctively 'British' or whether other cultures are, more or less, the same. Indeed, it may be that I am writing about the English Establishment, not the British ...
Hereabouts, the concept has a lot to do with an elite, a management class, usually based on inherited privilege, who are so self-evidently 'nice' that it is nasty and vulgar to speak against them; 'bad form' to suggest that they might be selfish or venal; shrill to question their motives.
One can earn approval by the Establishment by 'knowing their place', that is, accepting their greatness and one's own subservient position in their structures. I remember hearing one member of the servant class being referred to with great approval as "Salt of the Earth".
In the Church of England, Bishop Peter Ball was able to spend decades in the sadistic sexual abuse of young men, partly at least because he was a member of the Estabishment, with his inherited wealth, and the protection of Establishment circles reaching up, through archbishops, Public School head masters and High Court Judges, to Prince Charles.
I have never forgotten a passage in a TV play, The Other Man, subsequently converted into a novel, written by Giles Cooper in the early 1960s. Cooper is imagining a contrafactual Britain which, having done a disadvantageous deal with Hitler, is settling down to a subservient role within a Nazi Europe. In a particular military unit, Jewish officers are being filtered out and sent to labour camps. (George is a cadet member of the Establishment; Henry is a subversive.)
George moved over to the fireplace with his drink.
"So he's [a Jew] too".
"Jaffe? Well of course".
George smiled. "In his case I'm not surprised."
Henry's tone was level and angry as he asked why not.
"He's a first-class shit."
"You know, George,"said Henry, "you've a nasty shock coming to you one of these days when you suddenly find that all the nice people are wrong and all the shits are right." .
To be concluded.