Many people who have served in top government jobs do stick around, commenting favorably on how their former team is doing. Others, like the former Treasury secretaries Timothy F. Geithner and Steven Mnuchin, fade out of the limelight. Few remain as front and center as Mr. Summers, or as apolitical and provocative.

“He’s driven toward trying to find out what’s true rather than to give the politically correct answers,” said Ray Dalio, founder of Bridgewater Associates, the world’s largest hedge fund. “That’s caused him a lot of trouble, but I like it.”

There is a logical reason to stay in the mix. He has devoted his professional life to economic policy. There are also speaking fees.

“He’s takes pleasure in being not just a voice on issues, but also being a bit bombastic,” said Julia Coronado, founder of MacroPolicy Perspectives. She noted that doing so “serves him well economically”: Mr. Summers, like many academic economists, makes some of his money from consultancies and speeches, which tend to pay more for in-demand headline makers who have reputations as big thinkers.

Yet some blast Mr. Summers’s overheating crusade as an intellectual flip-flop.

After the Great Recession, Mr. Summers regularly acknowledged that the government’s spending response had been too meager. After that, he argued for years that the global economy was confronting a serious problem. Demand was too low, savings too high, and inflation and growth destined to be tepid — a phenomenon he called “secular stagnation,” borrowing from an economist who was prominent during the 1930s.

Stimulus stokes demand, and offered a hope out of the lackluster economic trap, so it seemed as if he should support it.

The “entire secular stagnation schtick has been turned on its head without explanation,” Mr. Hauser said.