A better question, though, is what are the Democratic bigwigs who are said to be “impatient” waiting for the great man to enter the presidential arena thinking? Imagine believing that what the Democratic Party is looking for in its leaders in 2019 is a handsy geriatric white man who distinguished himself over the course of a long political career by cozying up with the financial industry and bragging about being tough on crime in between cringe-worthy jokes about Strom Thurmond, at whose funeral he proudly delivered the eulogy (a fact unmentioned on his Wikipedia page).
During the Obama administration there was a creepy subgenre of liberal commentary involving Biden as a kind of sex symbol. This co-existed, in the way things always do when Democrats are in power, with a quiet recognition that his whole Crazy Uncle Joe act was a bit, uhh, much. The classic Onion parodies captured this ambiguity perfectly.
They are, fairly or not, how much of the public sees him. Maybe it was funny in 2012 to be the guy people casually make statutory rape jokes about, the same way that it was probably a good move in the ’90s to be the senator proudly announcing the fact that you want to punish 51 (!) different crimes with the death penalty. Who is the constituency for the latter these days, apart from Evelyn Waugh?