Mr. Biden, 76, has significant advantages, too. He is expected to post among the biggest fund-raising totals in the field when second quarter numbers are disclosed next month, and he enjoys widespread name recognition and good will among Democratic voters. Though he has spent much of his two-month-old campaign consumed with controversies tied to his decades-long career in government, he has so far maintained high poll numbers.

“Anything less than a dominant performance by a front-runner is always going to invite additional speculation,” said Fernand Amandi, a Democratic pollster based in Miami. “He has a little bit of a higher standard because of the legitimate concerns he himself acknowledges about his age.”

Mr. Biden has said it is fair to question his age and to judge him on his performance.

There is still plenty of time for him to improve his performance onstage — there is another debate scheduled for next month, and more after that — or for other candidates to surge.

The party divisions over how best to take on Mr. Trump were on vivid display at the debates Wednesday and Thursday, as Ms. Warren and Mr. Sanders led the charge for boldly progressive policy prescriptions that their allies hope could animate a Democratic base that was not sufficiently motivated in 2016. The centrists, led by Mr. Biden, advocated for more incremental change and a return to what they cast as pre-Trump-era normalcy.

Yet Mr. Biden, who has premised his entire candidacy on the idea that he is uniquely qualified to take on a president who is a threat to democracy, now faces pointed questions about his ability to do just that.

“If the contention is, he’s the one to go head-to-head, he’s going to have to kind of re-prove that,” said Sue Dvorsky, the former chairwoman of the Iowa Democratic Party. “Because last night put some cracks, I think, in that narrative.”

Ms. Harris, meanwhile, emerged as a clear winner from the debates. She spent part of the aftermath in the media “spin room,” where she entered to a cluster of boom mikes and the machine-gun staccato of camera shutters, and on Friday visited a detention center for migrant children in Homestead, Fla.