Gov. Ron DeSantis of Florida has run into a surprising buzz saw in his bid to sell himself as the Republican Party’s most electable standard-bearer in 2024 — and it has more to do with President Biden than it does with Donald J. Trump.

For months, Republican voters have consumed such a steady diet of clips of Mr. Biden stumbling, over words and sandbags, that they now see the 80-year-old Democratic incumbent as so frail that he would be beatable by practically any Republican — even a four-times-indicted former president who lost the last election.

As Mr. Trump’s rivals take the stage for the first debate of the 2024 primaries on Wednesday, the perceived weaknesses of Mr. Biden have undercut one of the core arguments that Mr. DeSantis and others have made from the start: that the party must turn the page on the past and move beyond Mr. Trump in order to win in 2024.

The focus on “electability” — the basic notion of which candidate has the best shot of winning a general election — was most intense in the aftermath of the disappointing 2022 midterms. Republicans were stung by losses of Trump-backed candidates in key swing states like Arizona, Michigan and Pennsylvania. And the issue offered a way to persuade a Republican electorate still very much in the thrall of Mr. Trump to consider throwing its lot in with a fresh face in 2024. It was a permission slip to move on.

But nine months later, interviews with pollsters, strategists, elected officials and Republican voters in early-voting states show that the dim Republican opinion of Mr. Biden’s mental faculties and political skills has complicated that case in deep and unexpected ways.

“I mean, I would hope anybody could beat Joe Biden at this point,” said Heather Hora, 52, as she waited in line for a photo with Mr. Trump at an Iowa Republican Party dinner, echoing a sentiment expressed in more than 30 interviews with Iowa Republicans in recent weeks.

Mr. Trump’s rivals are still pushing an electability case against the former president, but even their advisers and other strategists acknowledge that the diminished views of Mr. Biden have sapped the pressure voters once felt about the need to nominate someone new. When Republican primary voters in a recent New York Times/Siena College poll were asked which candidate was better able to beat Mr. Biden, 58 percent picked Mr. Trump, while 28 percent selected Mr. DeSantis.

“The perception that Biden is the weakest possible candidate has lowered the electability question in the calculus of primary voters,” said Josh Holmes, a Republican strategist and a longtime adviser to Senator Mitch McConnell, the Republican minority leader.

Though the urgency of electability has plainly waned, it remains one of the most powerful tools Mr. Trump’s rivals believe they have to peel the party away from him — and some privately hope that Mr. Trump’s growing legal jeopardy will eventually make the issue feel pressing again. For now, the fact that many polls show a razor-thin Biden-Trump contest has made it a tougher sell.

Conservative media, led by Fox News, has played a role in shaping G.O.P. views. Fox has often elevated Mr. DeSantis as the future of the Republican Party, coverage that has frustrated the former president. But the network’s persistent harping on Mr. Biden’s frailties may have inadvertently undercut any effort to build up Mr. DeSantis’s campaign.

More than two-thirds of Republicans who described Fox News or another conservative outlet as the single source they most often turned to for news thought Mr. Trump was better able to beat Mr. Biden in the Times/Siena College poll, a 40-point advantage over Mr. DeSantis. Those who cited mainstream news outlets also said Mr. Trump was the stronger candidate to beat Mr. Biden, though by less than half the margin.

There is little question that Mr. Biden has visibly aged. The president’s slip onstage at an Air Force graduation ceremony in June — his staff subsequently blamed a stray sandbag — is seen as a moment that particularly resonated for Republicans, cementing Mr. Biden’s image as frail, politically and otherwise.

Google records show search interest for “Biden old” peaking three times in 2023 — during his State of the Union address in February, when he announced his 2024 run in late April and when he fell onstage in June. The number of searches just for “Biden” was higher after his fall than it was around the time of his re-election kickoff.

Interviews with Republican voters in Iowa in recent weeks have revealed a consistent impression of Mr. Biden as weak and deteriorating.

“It’s just one gaffe after another,” Joanie Pellett, 55, a retiree in Decatur County, said of Mr. Biden as she settled into her seat in a beer hall at the Iowa State Fair four hours before Mr. Trump was set to speak.

“What strength as a candidate? Does he have any?” Rick Danowsky, a financial consultant who lives in Sigourney, Iowa, asked of Mr. Biden as he waited for Mr. DeSantis at a bar in downtown Des Moines earlier this month.

“He’s a train wreck,” said Jack Seward, 67, a county supervisor in Washington County, Iowa, who is considering whether to vote for Mr. Trump or Mr. DeSantis.

Kevin Munoz, a campaign spokesman for Mr. Biden, said Republican depictions of Mr. Biden as old were “recycled attacks” that had “repeatedly failed.”

“Put simply, it’s a losing strategy and they know it,” he said. “Republicans can argue with each other all they want about electability, but every one of them has embraced the losing MAGA agenda.”

Some Republicans worry that their voters have been lulled into a false sense of complacency about the challenge of beating a Democratic incumbent president. The last one to lose was Jimmy Carter more than four decades ago.

“Electability is more than just beating Biden — Republicans need to choose a candidate who can build a majority coalition, especially with independents, to win both the House and Senate,” said Dave Winston, a Republican pollster.

There were always structural challenges to running a primary campaign centered on electability. For more than a decade, Republican voters have tended to care little about which candidate political insiders have deemed to have the best shot at winning — and have tended to revolt against the preferences of the reviled party establishment.

Then there are the hurdles specific to Mr. Trump, who was portrayed as unelectable before he won in 2016, and whose 2020 loss has not been accepted by many in the party.

In a sign of how far electability has diminished, Republican voters today say they are more likely to support a candidate who agrees with them most on the issues over someone with the best chance to beat Mr. Biden, according to the Times/Siena College poll. They are prioritizing, in other words, policy positions over electability.

Mr. DeSantis has sharpened his own electability argument heading into the first debate, calling out Mr. Trump by name. “There’s nothing that the Democratic Party would like better than to relitigate all these things with Donald Trump,” Mr. DeSantis said in a recent radio interview. “That is a loser for us going forward as a party.”

The picture is brighter for Mr. DeSantis in Iowa, according to public polling and voter interviews, and that is where he is increasingly banking his candidacy. More than $3.5 million in television ads have aired from one anti-Trump group, Win It Back PAC. Those ads are explicitly aimed at undermining perceptions of Mr. Trump with voter testimonials of nervous former Trump supporters.

“For 2024, Trump is not the most electable candidate,” one said in a recent ad. “I don’t know if we can get him elected,” said another.

Likely Republican voters in Iowa see Mr. Trump as “able to beat Joe Biden” more than Mr. DeSantis despite that advertising onslaught, according to a separate Times/Siena College Iowa poll. But the margin is far smaller than in the national poll, and a larger share of Iowa Republicans say they would prioritize a candidate who could win.

Mr. DeSantis’s improved standing in the state when it comes to electability is heavily shaped by the views of college-educated Republicans. Among that group, Mr. DeSantis is seen as better able to beat Mr. Biden by a 14-point margin compared with Mr. Trump.

Mr. DeSantis faces his own electability headwinds. Some of those same party insiders who are worried about Mr. Trump topping the ticket have expressed concerns that the hard-line stances the governor has taken — especially signing a six-week abortion ban — could repel independent voters.

Mr. Danowsky, the financial consultant who was at the bar in downtown Des Moines, worried that Mr. DeSantis was “a little extreme,” including on transgender rights.

But more Iowa Republicans volunteered concerns about Mr. Trump’s viability as the top reason to move on from him, even as they saw Mr. Biden as weak.

“I might be one out of 1,000, but I don’t think he can beat Biden,” Mike Farwell, 66, a retired construction worker in Indianola, said of Mr. Trump. He added that Mr. Biden “would be an easy president right now to beat” if he faced a strong enough opponent.

Don Beebout, 74, a retiree who lives in Chariton and manages a farm, was worried about Mr. Trump as the party nominee as he waited to hear Mr. DeSantis speak at the state fair. But he also was not sold on any particular alternative.

“He may be easy to beat,” he said of Mr. Biden, “if we get the right candidate.”

Maggie Haberman and Jonathan Swan contributed reporting.