Boris Johnson aside, their public perception or popularity seems way down their list of priority's. Mogg briefly enjoyed riding a bit of a crest when he became an Alt-Right icon for a while but when it comes down to it as long as they have their large bank balance and perks they don't seem too arsed about people liking them. Boris craves popularity and adoration and I do think he's struggling with the idea of not being the lovable rouge anymore but the rest of them couldn't give a shit if the general public don't like them. Obviously this presents a problem for them when it comes to voting time but they can rely on the electorate being a bit thick, on the whole.
Alex does seem genuinely puzzled as to why Keeves doesn't love him, doesn't he? It's worth finding Otto English's pieces on Johnson in Byline Times detailing the giveaway phrases used to describe non-Etonians. The tossers really do believe they have a divine right to rule over us.