Decades later, he enjoys far more good will from Democratic voters after having served as vice president, Mr. Lux said. “I don’t think it has anything to do with his age,” he said. “I just think that’s him. I don’t think that’s slipping in any sense of the word.”

If elected, Mr. Biden, who is 76, would be the oldest president in history at his inauguration, though Senator Bernie Sanders of Vermont, one of his chief rivals for the Democratic nomination, is older, at 77. President Trump, who has faced questions about his mental fitness, is 73.

“You can definitely tell his age,” said Bryce Smith, the Democratic chairman in Dallas County, Iowa, when asked about Mr. Biden’s misstatements, even as he also noted that “no one is perfect” and nodded to the former vice president’s “lifetime of service.”

“As a 27-year-old,” Mr. Smith added, “it’s harder for me to relate to how he talks, per se, than it is to relate to someone who’s half his age running for president. It doesn’t mean he’s wrong.”

Mr. Biden’s allies note that despite facing questions about his age and his propensity for gaffes all summer, Mr. Biden remains the early Democratic favorite, at least in most polls. Reporting on the ground in Iowa, however, suggests signs of an enthusiasm challenge.

“Time and again, the press fixates on supposed Joe Biden ‘gaffes’ — and time and again, voters say they don’t care,” Andrew Bates, a spokesman for Mr. Biden’s campaign, said.

David Axelrod, who served as former President Barack Obama’s chief strategist, said the episode of the fumbled war story “reflects both the lack of discipline and the extraordinary empathy” of Mr. Biden.

“In this story you have the risk and the strength of Biden,” Mr. Axelrod said. “The risk being that he is a gaffe-prone guy, and we’ve seen that throughout his career. I don’t know whether it’s better or worse now, but it’s a factor, and at his age people are going to pay more attention to each of them. But on the other hand, he projects extraordinary empathy, and that empathy is a huge strength.”