Re 'unelected' PMs, the big difference between the Brown instance and the May one is that, back then, there were right-wing nutters dressed up as BEER-bottles following Brown around, so that they could shout "Bottler Brown!" at him. In other words, they branded him as a coward.
I don't notice any left-wingers doing that to May. When she gained the Tory 'crown', I tried to think of a suitably alliterative descriptor for the lady. The best the thesaurusi could come up with was 'milksop'. If we lefties weren't so polite, we migh be seeing such people dressed as MILK-bottles yelling, "Milksop May!" at her!