Washington — In the wake of President Biden’s calamitous debate performance last week, his campaign unveiled a new ad Monday highlighting his more energetic appearances after the debate, in which he called out former President Donald Trump for the many lies and exaggerations he told during the debate. Those post-debate appearances largely took place in front of friendly crowds — much smaller than the 51 million who saw the debate — and in at least two of those appearances, the president read his remarks off a teleprompter. 

After the debate, which has prompted concern among Democrats about President Biden’s fitness for a second term, his family is encouraging him to stay in the race and keep fighting, a source with knowledge of the conversations told CBS News. The president was with family on Sunday for a long-planned gathering at Camp David. White House and campaign officials also say there are no plans to make significant changes to the president’s campaign schedule or cadence in the wake of the debate. 

Mr. Biden’s showing at last week’s debate, where the 81-year-old president struggled to correct falsehoods from Trump, while delivering rambling answers to key questions in a hoarse voice, has prompted concerns about his ability to serve and suggestions from some Democrats that he should reconsider his presidential bid. The New York Times editorial board explicitly called on him to drop out of the race. But in the aftermath of the debate, his campaign has said that there are no conversations about the president leaving the race. 

One of the family members urging Biden to stay in the race is the president’s son, Hunter Biden, two sources told CBS News. The sources say a few family members are frustrated about how Mr. Biden was prepared for the debate, which took place at Camp David in the days ahead of Thursday’s matchup. Still, a campaign aide stressed that reports of conversations about the race among family members are the result of “informal” family chatter, making clear that the sentiments did not come as part of a formal family discussion. 

“The trip was planned. Of course it came up. But it was not a ‘family meeting’ or anything like that,” an aide said. Other aides said the family had gathered for a photoshoot with Annie Leibovitz, who previously photographed the Obama family. The weekend was selected for the photoshoot because the family typically gathers for the July 4 holiday.

Another source close to first lady Jill Biden said she “does not feel the way” that some members of her family are being described in reports, which outline concern among family members about how three key advisers — White House senior adviser Anita Dunn, her husband, personal counsel Bob Bauer and former White House chief of staff Ron Klain — led debate preparations. 

Another source blamed Mr. Biden’s inner circle for the debate preparations, telling CBS News that instead of allowing the president to be himself, “they pumped him up with facts that no one cares about” to fact check former President Donald Trump, and exhausted him. This source also blamed CNN moderators Jake Tapper and Dana Bash for not fact-checking Trump. Before the debate, CNN said that Tapper and Bash would not try to fact-check the statements made by Trump or Mr. Biden in real time, although CNN’s political director, David Chalian said they had the latitude to do so if they deemed it to be necessary, according to the Associated Press. 

“Obviously, if there is some egregious fact that needs to be checked or the record needs to be made clear, Jake and Dana can do that,” Chalian said before the debate. “But that’s not their role. They are not here to participate in this debate. They are here to facilitate a debate between Trump and Biden.”

The first lady, who is the subject of Vogue’s August cover, spoke with Vogue over the weekend from Camp David following the debate, telling the magazine that they “will not let those 90 minutes define the four years he’s been president. We will continue to fight.” 

Top officials spent the weekend holding “honest conversations” with Democratic lawmakers, donors and other stakeholders. Those officials steadfastly insist that Mr. Biden is in the best position to beat Trump. Another source told CBS News that despite the president’s poor debate showing, there’s been an uptick in fundraising and volunteer engagement, claiming that the campaign has received hundreds of new resumes and new donor interest. 

“There has been a real jolt of energy,” the source said.  

But for voters, the debate came at a time when many Americans believe the president may be too old for the job. And a new CBS News poll found that since the debate, the percentage of voters who say that Mr. Biden has the cognitive ability to serve as president has dropped from 35% just weeks ago to 27%. Now, nearly three-quarters of voters say the president shouldn’t be running for reelection, including almost half of Democrats, according to the poll. 

Still, views of former President Donald Trump’s cognitive ability aren’t far off. About half of all voters say Trump doesn’t have the cognitive health to serve. But voters generally agree that Trump was stronger during the debate, presenting his ideas more clearly while appearing more confident and presidential.

Robert Costa contributed reporting.

Nancy Cordes

Nancy Cordes is CBS News’ chief White House correspondent based in Washington, D.C. Cordes has won numerous awards for her reporting, including multiple Emmys, Edward R. Murrow awards, and an Alfred I. duPont-Columbia University Award.