Not essential. Churchill went to Harrow !
What is disturbing is not that this Old Etonian holds high office, nor that men who went to good public schools do (e.g.the Deputy Prime Minister was at Westminster School), but that so many of his cabinet went to the same public school. On what basis were they chosen by him? It looks more than a coincidence and smacks of a certain narrowness of mind, a thinking that a fellow Etonian is the best man simply because of that fact
In itself, that people who have had the best education end up in top jobs to a disproportionate degree, is rather more of an indictment of the state schools of their time than of anything else. That's not to ignore that public schools are affordable by the rich, not the ordinary, families, that money helps in many careers and that the rich parents are well-connected and their children have a better chance of advancement by use of those connections, perhaps made or ensured by their children being at the same public schools.