How this local shift in perception could affect his current political prospects is unclear — New York is hardly known for swaying presidential elections. But it suggests that opinions of Michael Bloomberg are mutable — in some part because he is many things at once but also because his presentation lacks either the emotion or the phoniness to inspire visceral feelings one way or the other.

As it happened, on the day of the debate The Times published an opinion piece by Shira A. Scheindlin, the former federal judge who presided over the 2013 lawsuit that effectively ended the city’s stop-and-frisk program, ruling that the tactics violated the constitutional rights of people of color. As mayor, Mr. Bloomberg fought the suit after aggressively expanding the protocols. He apologized for the racist policy only in advance of his entry in the presidential race.

But Ms. Sheindlin was not writing to excoriate him; to the contrary, she wanted the world to know that she did not think that he was racist. She then cited the many opportunities Mr. Bloomberg created for minorities with a clarity the candidate himself was unable to summon at the debate.

In retrospect, it is easy to imagine that Mr. Bloomberg and his police captains were the only ones slow to recognize the indignity of halting the free movement of young black and brown men — first on the premise that they might be carrying guns, and then on the basis that they were walking around with weed, inexplicably viewed as equally threatening to the social order.

In an August 2012 New York Times poll, though, at a point when close to 330,000 stops had already been recorded that year, a higher percentage of respondents said they found the practice of stop-and-frisk “acceptable’’ than those who found it “excessive.” Even after racially dubious video clips of Mr. Bloomberg emerged recently, he somehow polled significantly higher with African-Americans — and women — than Elizabeth Warren did.

The same poll, of Democrats and independents inclined toward Democratic candidates, suggests that the Trump presidency has not expunged a taste for billionaires among working people. Of those making less than $50,000 and of those without college educations, more favored Mr. Bloomberg than any other candidate in the Democratic field save for Bernie Sanders.

The great intrigue of a general election race between the self-made billionaire and a self-proclaimed one is the promise of a matchup between two competing strains of vanity, one born of extreme confidence and the other of enduring insecurity. It’s still possible we’ll get to witness it.