“Since 2008, governments have managed to get many of their headline indicators to start moving in the right direction,” Mr. Davies said in a phone interview, referring to economic figures like a rising gross domestic product and falling unemployment, “but often those indicators are masking all sorts of underlying problems. If the politicians had been a bit more sensitive to some of the underlying suffering and the emotional aspects of the economy, maybe the ruptures of 2016 were things they would have seen coming.”

Mr. O’Brien often expresses great sympathy for Leavers, many of whom he believes have been lied to by political opportunists. That said, his show is hardly a venue for national reconciliation. His audience is mostly Remainers; many tune in to eavesdrop on the public eviscerations of Leavers.

“I wouldn’t be surprised if many people, especially if they don’t agree with him to start with, find O’Brien’s style of argumentation grating,” said Hugo Mercier, a co-author of “The Enigma of Reason: A New Theory of Human Understanding.” “He keeps interrupting, and sometimes starts arguments over seemingly noncrucial points. As much as I might understand his anger and frustration, this is likely counterproductive.”

Mr. O’Brien path to a job in the media ran, improbably enough, through haberdashery. Raised in Kidderminster, a small town not far from Birmingham, he worked at Aquascutum, a luxury men’s shop in London, after college. One day in 1996, some of his colleagues returned from 10 Downing Street with material for a white suit for the prime minister, John Major. The color was so out of character that Mr. O’Brien called the gossip editor at The Daily Express and pitched it as an item.

“The guy said, ‘I’ll give you 500 quid,’” Mr. O’Brien recalled. “I said, ‘Can I have some shifts instead?’”

He started writing showbiz column items for a variety of London tabloids. His first radio job came years later, and his breakout moment occurred in 2014, in a contentious LBC interview with the future Brexiteer Nigel Farage. (The clash has been viewed on YouTube nearly one million times.) Mr. Farage was the leader of the U.K. Independence Party, known as UKIP, and had been accused of xenophobia.

“You’ve mentioned your discomfort at listening to foreign languages on a train recently,” Mr. O’Brien said.