Bloomberg's always a good example of what you mentioned in my opinion, socially progressive fiscally moderate-to-conservative billionaire tycoon whose big personal issues are gun control and salt. Yeah, that's the kind of guy who the country is clamoring for. But in punditocracy he's kinda the king of the "important people" circle. Even Trump figured out to run on immigration/trade/jobs while working his wealth both ways "higher taxes/I'm awesome!" (As did Perot.)
Rudy had something similar going in GOP circles where they imagined all this crossover appeal because he was a R who won in NYC and pro-choice, pro-gays, etc. yet was tough on crime/terrorism. Colin Powell before him only with foreign policy instead of crime/terrorism. Sam Nunn an inverse.
Obama, Bill Clinton, and H.W. Bush were kinda these types of Presidents, although not candidates perhaps, with socially acceptable views (even if wink-wink) and otherwise not too much of a boat rocker. W. Bush by contrast had that whole born again thing going on and went to bed early. Reagan had his unacceptable rhetoric. Carter had his scolding and being his own chief of staff thing, he also didn't care for the gala circuit either.