HOUSTON — Joe Biden has a record that spans nearly half a century, with some positions that can easily be cast as out of touch with today’s Democratic voters. He makes near-daily verbal miscues about matters trivial and weighty that can often undermine his own campaign.

Yet after Thursday’s third debate, the former vice president’s standing in the race appears as durable as ever, with none of his opponents quite sure how to attack him. None of their efforts so far have reshaped the race, and on Friday, Democratic candidates waded into a delicate discussion about Biden’s age and mental acuity that could just as easily backfire on those raising the questions as harm the 76-year-old Biden.

Seeking to deflect concerns, Biden responded on Friday by committing to release his medical records within the next several months, before Democratic primary voters begin selecting their nominee. Then, as a joking profession of strength, he challenged a reporter to wrestle. Biden had surgery three decades ago to repair two brain aneurysms, and records released in 2008 showed he made a full recovery.

“Before there’s a first vote, I’ll release my medical records,” Biden said, adding that he will do so after his next physical. “There’s no reason for me not to release my medical records.”

Releasing his medical records could further insulate him from questions about his age, a topic that his rivals have started raising more prominently in recent days. Two other candidates are in their 70s; an aide to Sen. Elizabeth Warren (D-Mass.) said late Friday that Warren, 70, also would release her records before the first voting. Sen. Bernie Sanders (I-Vt.), who is 78, has not yet fully committed to releasing his.

For Biden and the party he seeks to lead, questions about his stamina and fitness strike at the heart of the theory he has pushed to voters, so far successfully: that he would be the nominee best equipped to battle President Trump in the general election. Any physical vulnerability would threaten to undermine that perception of electability, a voter viewpoint that has frustrated his Democratic opponents. President Trump has sought to raise questions about Biden’s vitality by characterizing him as “Sleepy Joe.”

The debate and its aftermath marked a new phase for the campaign, as it heads into a crucial stretch this fall, with what has felt like a rollicking and unpredictable primary race beginning to stabilize.

“Joe Biden went into the debate as the front-runner and Joe Biden came out of the debate as the front-runner,” said Terry McAuliffe, the former Virginia governor and a onetime party chairman who is neutral in the race.

The numerous attacks on Biden have not landed, he said, because “we all know Joe. Joe Biden has been in this a very long time. People love Joe Biden. And honestly, people don’t want to see them attacking President Obama’s record.”

The other candidates also have attempted a delicate dance: praising President Barack Obama while trying to criticize Biden’s record within the administration. But so far Biden has easily dispatched those critiques by hugging closely the most popular figure in the party.

Sen. Kamala D. Harris (D-
Calif.), who in the first debate launched the most biting attack on Biden for his decades-long opposition to mandated busing as a way to integrate schools, has since recalibrated to focus her attention largely on Trump. Sen. Cory Booker (D-N.J.) has attempted to ridicule Biden for a crime bill that led to mass incarceration, while Sen. Kirsten Gillibrand (D-N.Y.) said he was weak on women’s rights. Neither of the first two attacks boosted the senators; Gillibrand left the race soon after her critique.

During Thursday’s debate, former Obama administration housing secretary Julián Castro made an aggressive attempt to highlight Biden’s mental acumen by repeatedly suggesting that he had forgotten something that he’d said just two minutes earlier. But Castro was off on his facts, stifling the potential damage he could inflict, even as other candidates began raising similar concerns.

“I think that we are at a tough point right now, because there’s a lot of people concerned about Joe Biden’s ability to carry the ball all the way across the end line without fumbling,” Booker said during a CNN interview after the debate, adding that Castro raised “really legitimate concerns.”

While Booker said he did not think Biden was too old to be president, he said, “There are definitely moments when you listen to Joe Biden and you just wonder.”

Asked Friday in Houston how he can prove his abilities to rivals — and voters — Biden replied, “I carry the ball over the finish line.” But he also said that questions about the impact of his age were fair game.

Other candidates were squeamish about the direction of the discussion, with Sen. Amy Klobuchar (D-Minn.) calling Castro’s criticism “so personal and so unnecessary” and former congressman Beto O’Rourke (D-Tex.) saying, “That kind of personal attack, I don’t think, is what we need right now and is insufficient to the challenges we face.”

“I thought Castro’s attacks were ludicrous,” McAuliffe said. “It’s not helpful to the party to have Democrats chewing on each other this way. The thing I worry about is when people watched that and saw it they probably turned it off and said, ‘I’m sick of this.’ We need a positive message.”

Castro sent out a fundraising email Friday, saying he was being “viciously attacked.”

“I could either play it safe and give Vice President Biden a free pass like everyone else,” he wrote. “Or I could speak up.”

Biden campaign advisers said Castro’s attack was “a cheap shot” and pointed out that past attacks on him have not worked out for the attacker.

The Biden campaign acknowledges that Biden misspeaks with frequency, and aides attempt to dispel the idea that it is a result of age by pointing to a string of gaffes that goes back decades. They have viewed it as a quality that gives him some measure of authenticity, with one adviser saying, “The biggest mistake we could make as a campaign is to not let Joe Biden be Joe Biden.”

“I think the gaffes are already baked into the cake. Democrats know his flaws — but right or wrong — the belief remains for now that he’s the person on the stage who can win,” said Glen Caplin, a Democratic strategist who was a top adviser to Gillibrand’s campaign. “And that right now is the altar a lot of people are praying at. Until that core argument of his campaign — he is the electable one — is shaken, the dynamic of the race will probably be the dynamic of the race for a while.”

Ironically, considering Trump’s criticisms of the former vice president, the parameters of the Democratic race are reminiscent of the Republican primary campaign in 2016, during which rival campaigns assumed to their detriment that Donald Trump was flawed and would eventually collapse on his own.

Biden’s performance in the debate reminded voters of the promise and peril of his candidacy. He was strong in its early portions, eager to defend his health-care plan and feisty as he pointed toward the high costs associated with the Medicare-for-all proposal pushed by Sanders and Warren. Biden also closed the evening by retelling the emotional story about the death of his wife and daughter in a car accident in 1972, and the death of his son in 2015.

But he still made misstatements (“Nobody should be in jail for a nonviolent crime,” he said, when he meant nonviolent drug offenses). He meandered on several answers, quickly jumping around to various unrelated topics.

Biden’s clumsy way of talking about race also was highlighted.

He was asked on Thursday night to respond to a comment he made in 1975, which The Post resurrected earlier this year, when he spoke about segregation by saying, “I don’t feel responsible for the sins of my father and grandfather. I feel responsible for what the situation is today, for the sins of my own generation. And I’ll be damned if I feel responsible to pay for what happened 300 years ago.”

Biden began his response by talking about the need to increase teacher salaries, and to hire more school psychologists who can meet needs that aren’t being met at home. He then said social workers are needed in some cases because poor parents need help raising their children.

“They don’t know quite what to do,” Biden said. “Play the radio, make sure the television — excuse me, make sure you have the record player — on at night, make sure that kids hear words. A kid coming from a very poor school — a very poor background — will hear 4 million words fewer spoken by the time they get there.”

Given that the question was about repairing the legacy of slavery, some viewed Biden’s answer as being critical of black parents.

He then transitioned away from education to talk about Venezuela, how he knew and had confronted President Nicolás Maduro, and had tried to improve conditions in Latin America.

The next speaker suggested he was at a loss to respond.

“Well,” Castro said. “That’s quite a lot.”

Annie Linskey contributed to this report.