Bill Murphy, an 80-year-old retired veterinarian in suburban Phoenix, sometimes blanks on names he could once summon with ease, so he has empathy for 81-year-old President Biden. But he winced when he watched Mr. Biden defend his mental sharpness at a news conference, only to mix up the presidents of Egypt and Mexico. Mr. Murphy, a Republican, believes Mr. Biden is not up to another term.

Mary Meyer, an 83-year-old avid hiker and traveler who lives in the high desert north of Phoenix, took issue with a special counsel’s report that characterized him as elderly and forgetful — a similar assumption that strangers at the supermarket sometimes make about her capabilities.

“I look at him as a peer,” said Ms. Meyer, who plans to vote for Mr. Biden. “I know what he’s capable of. I know it’s not as bad as everybody thinks.”

To voters in their 70s and 80s, the renewed questions swirling around Mr. Biden’s age and fitness resonated in deeply personal ways. The special counsel report cleared him of criminal charges in his handling of classified documents but described him as a “sympathetic, well-meaning, elderly man with a poor memory.”

Some of Mr. Biden’s generational peers and supporters insisted the characterization was nothing more than a calculated political ploy to undercut his campaign, and play on perceived weakness. Many noted their own vibrant and busy lives, filled with mental and physical activity.

The criticism of Mr. Biden as forgetful and incapable of serving echoed slights and discrimination they had felt. Others thought of their own struggles as they hit their 80s, and questioned any 80-year-old’s ability to lead the nation.

They said they knew what it felt like to be drained after a two-hour drive, to struggle to recall names and facts that once came easily, to feel wary climbing up a ladder or mounting a bicycle. Republican and Democratic voters alike said they could not help but empathize with Mr. Biden’s verbal slips, and with the crush of judgment that followed them.

Despite efforts from the White House to discredit the portrait of Mr. Biden as forgetful, some voters worried about how a president their age would, in the years to come, continue to respond to the stress of wars in the Middle East and Ukraine, handle the toil of campaign rallies or endure draining overnight flights to summits around the world.

“What’s it going to be like when he gets a call in the middle of the night that there’s been a bombing, or there’s been another shooting?” asked Jan Kallheim, 83, a retired nurse from Mankato, Minn., who spends her winters in Arizona and describes herself as a conservative.

Surveys show that many voters share her concerns. In a New York Times/Siena poll of six battleground states last fall, more than 70 percent of voters agreed with a statement that Mr. Biden, 81, is too old to be an effective president, though voters 65 and over were slightly less likely to judge him as too old. More than 60 percent of all voters polled did not think he had the mental sharpness to be president. On that question, voters 65 and older were evenly divided.

On Friday, The Times talked to nearly two dozen older voters about their thoughts on age and the presidency, and received written responses on the topic from around 150 more. The responses, while not scientific, seemed to show that few of Mr. Biden’s supporters were swayed by the special counsel report, or his performance at the White House press conference responding to it. Those who worried about Mr. Biden’s age had already felt concern.

Many of the older Democrats and independent voters who responded said they were more perturbed about Mr. Trump’s temperament than about Mr. Biden’s age.

On Friday, Ms. Kallheim and a group of golf-loving, mah-jongg-playing, mostly conservative friends from the Midwest were finishing dinner at a Panera in the Sun City area, a suburban Phoenix mecca for retirees and snowbirds. There, residents zip around in golf carts from tee times to woodworking courses to the lap pool.

The friends saw the upcoming election as a distressing choice between what they called “grandpa and crazy.” They said they would rather write in Republicans Nikki Haley or Liz Cheney than vote for Mr. Biden or Mr. Trump.

Asked whether they wanted younger alternatives, all said yes.

“You don’t bounce back,” said Ms. Kallheim, who tends to vote Republican but said she would never vote for Mr. Trump. “I don’t care how physically active you are, if you take pills or not — your acuity at 80 isn’t as good as it is at 60. There is deterioration.”

“I don’t think like I did when I was younger,” said Lorene Bleess, 85, who recently had to give up pickleball after an illness.

But other voters in their 70s and 80s said they still felt as mentally agile as they did in their 50s and 60s, and had few concerns about Mr. Biden’s age. They pointed out that politicians of all ages — including Mr. Trump — make verbal flubs and forget names and dates.

“I have been in a position where I was asked something on the spot and I have stumbled here and there,” said Beverly Edmond, 74, a Democrat and retired university administrator in Lithonia, Ga. “It happens, but it is not a reflection of intellect or capability. This has been blown out of proportion.”

To underscore that point, after the special counsel’s unflattering description of Mr. Biden, Democrats circulated videos showing Mr. Trump, mixing up Ms. Haley with Nancy Pelosi, and Mike Johnson, 52, the speaker of the House, mixing up Israel and Iran.

Some older voters said it was unfair to judge Mr. Biden solely by his age. But in an angrily polarized and youth-obsessed culture, they were not surprised to see commentators dismiss Mr. Biden as “doddering” and “out to lunch,” or so many voters deem him “just too old” in opinion surveys.

Ann Marie Cunningham, 86, a retired teacher and social worker in Chicago, said that some criticism of Mr. Biden’s age flowed from a widespread bias against older people. When she was working at a nursing home in her early 70s, she said a new manager pushed out many of the older workers, including her.

“Being older is a liability,” Ms. Cunningham said.

She uses a walker to get around her high rise but says that her mind is as clear as ever. Ms. Cunningham, a registered Democrat who considers herself an independent, has marveled at Mr. Biden’s physical fitness well into his 80s.

“He rides a bike!” she said. “I couldn’t ride a bike if my life depended on it.”

Still, she has noticed his verbal lapses. When Ms. Cunningham was a teacher at a Catholic school in Chicago for 35 years, she had students who suffered from a stutter, like Mr. Biden.

“Sometimes he’ll forget a word or get things garbled,” she said, which she believes has to do with his early speech difficulties. “That’s not easy.”

Mr. Biden was not Ms. Cunningham’s first choice for president. She voted for him, with a little reluctance, but has been satisfied with his achievements in office, especially lowering costs on prescriptions for older people, and said she would vote for him again.

But Harry W. Hepburn III, an 82-year-old clock repairman in Harrison, Maine, sees Mr. Biden’s age as a risk that imperils the country. Mr. Hepburn has empathy for the slip-ups that come with age and gets bothered when he occasionally forgets a name, “when things used to come to me so quick, like boom-boom-boom.”

A registered Republican, Mr. Hepburn voted for Mr. Trump in 2016 and 2020 and plans to vote for him again in November because of his efforts to restrict immigration and his business sense, he said. He does not think age has yet taken a toll on Mr. Trump.

Mr. Hepburn stays active by tossing a football with his grandchildren and still shovels snow off his porch roof. He said he feels exceptional for 82, but doesn’t believe the same is true of Mr. Biden.

“I think he has lost his ability to think on his feet,” Mr. Hepburn said, while continuing to tinker with a clock in his workshop on Friday afternoon. “He scares me. Watch how he walks — he walks like a guy who doesn’t have it anymore.” (The White House report of Mr. Biden’s 2023 physical described him as a “healthy, vigorous 80-year-old man.”)

Even some Democratic supporters, like Sarah Shankman, an 80-year-old novelist in Santa Rosa, Calif., say they wish Mr. Biden would leave the stage. Ms. Shankman compared Mr. Biden’s re-election bid with how Justice Ruth Bader Ginsburg resisted calls to retire from the Supreme Court during the Obama administration, only to die and be replaced by a Trump nominee.

“I think his heart is in the right place, but I think his ego has gotten in the way,” she said. “I think Biden is having a very difficult time just coming to face the inevitable.”

Opinion surveys suggest Mr. Biden’s older supporters have stuck with him even as he has lost support among young voters upset about issues like the war in Gaza.

“We all have deficits as a young person and deficits as we age,” said Linda Georgeson, a 74-year-old retiree and Democrat, of Bayfield, Wis.

As a court administrator, Ms. Georgeson ruled on cases involving competency and abuse of older people, and came away believing that age does not necessarily determine mental acuity.

After reading the special counsel’s report, Ms. Georgeson felt insulted “on behalf of all older people,” she said. “I think it’s time in this country we understood everybody’s going to age.”

Halina Bennet contributed reporting.