Former Housing Secretary Julian Castro called out the elephant in the room when he questioned whether Joseph R. Biden was mentally fit to be president, pulling back the curtain on a debate that has been raging inside the Democratic Party since the former vice president entered the race.

For Carlos Cardona, chairman of the Laconia New Hampshire Democrats, it was an uncomfortable subject that nevertheless had to be broached.

“He is like our Uncle Joe. People love Joe. Joe is a good person,” Mr. Cardona said. “It is just more and more we are seeing a different Joe, and when you love somebody and they are changing, mentally or physically, and they are aging in a way you are not excited about, it is bittersweet.”

Mr. Cardona was among the 14 million viewers that tuned into the third presidential debate and watched Mr. Castro passionately challenge Mr. Biden to come clean on whether he was having memory lapses on stage.

The line of attack generated groans from the audience and an immediate backlash from talking heads in the mainstream media — and even some of the other Democratic 2020 presidential hopefuls — who accused Mr. Castro of playing the age card against Mr. Biden, 76.

But Mr. Cardona, who is backing 78-year-old Sen. Bernard Sanders, said he doesn’t didn’t see it that way.

“I know plenty of 70-year-olds who can run a marathon better than me, and probably outscore me on an SAT test,” he said. “I don’t play the age game. I think it is about the capability [of a candidate] for me more than anything.”

Sen. Cory A. Booker of New Jersey raised similar concerns in a post-debate interview, telling CNN “there are definitely moments when you listen to Joe Biden and you just wonder.”

But Sen. Amy Klobuchar of Minnesota took Mr. Biden’s side, invoking the “T-word” against Mr. Castro. “When a statement like that is made … it feels like something Donald Trump might tweet out,” she said.

For his part, Mr. Biden is making light of the situation. Asked over the weekend whether he would disclose his medical records, he said, “You wanna wrestle?”

Mr. Biden’s mental fitness has been a big question mark hanging over the race.

President Trump has tried to stoke the uncertainty by nicknaming him “sleepy Joe,” and describing him as “crazy” and a “low IQ individual.”

While nobody suggested that Mr. Biden is feeble or suffering from dementia, the most common assessment was that he is not at the top of his game. His performance in the debates this year were a mere shadow of his debate stage prowess in years past, such as with Republican Paul Ryan in the 2012 vice presidential debate. After the Ryan bout, The Guardian described Mr. Biden as putting on an “alpha-male display,” and Rolling Stone magazine said he “steamrolled” Mr. Ryan.

None of those descriptions are being applied to Mr. Biden’s performances these days. After last week’s debate, The Washington Post graciously gave him a B+.

Jim McLaughlin, a Republican pollster working with the Trump reelection campaign, said they were not polling about Mr. Biden’s mental acuity, but Democrats and news organizations were asking the question.

“I’m sure it is already showing up in surveys, and I’m sure it is being asked,” he said. “It is going to come. It’s out there.”

He said questions about age and fitness for a successful campaign cut to the heart of Mr. Biden’s chief argument for his candidacy: that he is the candidate best able to defeat Mr. Trump.

“That’s his message: ‘I am the guy who can beat Trump.’ But I don’t think at the end that’s going to be enough for the Democratic primary voters,” Mr. McLaughlin said.

Mr. Biden has suffered from a series of self-inflicted wounds. There have been verbal gaffes, well-documented memory woes on the stump, and recently blood pooling in his eye during a town hall-style forum on CNN. Things got tougher in the debate. His teeth appeared to almost shake loose. Mr. Castro passionately said, “Are you forgetting what you said just two minutes ago?”

The Biden camp later correctly pointed out that Mr. Castro had misconstrued the former veep’s comments.

Still, the clip of the exchange went viral and dominated debate coverage, forcing Democrats to ponder whether Mr. Biden is the best equipped to not only defeat Mr. Trump but also lead the nation.

Thom Hart, former chairman of the Scott County Iowa Democrats, said he thought Mr. Castro’s attack smacked of desperation from a long-shot candidate.

He predicted it won’t have a lasting impact on the race, comparing it to when Sen. Kamala D. Harris of California jabbed at Mr. Biden in the June debate for opposing federal busing to desegregate public schools in the 1970s.

The feisty confrontation gave Ms. Harris a temporary bounce in the polls but didn’t knock Mr. Biden from the front-runner position or alter the long-term shape of the race.

“The more minor folks try to have their come-out moment by peeling one of the front-runners, and I think people are not real keen on that Trumpian approach to the nomination,” Mr. Hart said.

What’s more, he said people are missing Mr. Biden’s meandering style.

“I have known Biden for 40 years, and I think that has always been part of his makeup,” Mr. Hart said. “I would put him kind of in the category of a smart-aleck who is good with the quips, and I don’t see that as mental acuity thing. I think that is the personality and I think a lot of people are really focusing on that. I also think if that were a disqualifier Trump would not be president.”

Mr. Castro on Sunday lost one of his highest-profile endorsement with Rep. Vicente Gonzalez saying he is switching his support to Mr. Biden, though the Texas Democrat said his switch wasn’t a direct result of, or backlash from, the Thursday night debate moment.

Mr. Gonzalez, who endorsed Mr. Castro back in March, said the former San Antonio mayor wasn’t performing well in the polls.

“I think at this point in time we need to narrow the field and unite as Democrats to defeat Trump in November 2020,” he said on CNN’s “State of the Union.”

With Mr. Gonzalez gone, Mr. Castro now has two endorsements by sitting members of Congress, one of them being twin brother Joaquin Castro. The other is Rep. Colin Allred of Texas.

• Gabriella Muñoz contributed to this report.

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