As the field was being set Wednesday for the next debate, the Democratic National Committee still struggled to convince the excluded candidates and their followers that it wasn’t putting its finger on the scale as it did in the 2016 primary race.

Ten candidates appeared to have qualified for the third debate ahead of Wednesday night’s deadline, triggering a fresh round of scrutiny over the DNC’s qualifying criteria and leaving those relegated to the sidelines on death watch and leading Sen. Kirsten E. Gillibrand of New York to call it quits on her campaign.

So far, 10 candidates appear to have made the cut for the Sept. 12 debate in Houston, including former Vice President Joseph R. Biden and five sitting senators — Sens. Bernard Sanders of Vermont; Elizabeth Warren of Massachusetts; Kamala D. Harris of California; Cory A. Booker of New Jersey; and Amy Klobuchar of Minnesota.

The other four who have met both the fundraising and polling thresholds are South Bend, Indiana, Mayor Pete Buttigieg; former Rep. Beto O’Rourke of Texas; entrepreneur Andrew Yang; and former Housing Secretary Julian Castro.

The prospect of a 10-candidate, single-night event would be a change from the 20 candidates who participated in the first two debates, which each took place over two nights.

It also marked another likely turning point in the campaign, coming days after a pair of candidates — Washington Gov. Jay Inslee and Rep. Seth Moulton of Massachusetts — pulled the plug on their underdog bids, raising the number of candidate dropouts to five since July.

Ms. Gillibrand, who had cast herself as a warrior on feminist issues, added her name to that list Wednesday night after it became clear she would not make the debate stage.

“After more than eight incredible months, I am ending my presidential campaign,” Ms. Gillibrand said in a video posted on Twitter. “I know this isn’t the result we wanted. We wanted to win this race, but it’s important to know when it’s not your time.”

Ms. Gillibrand said she would work to unite the party behind defeating President Trump in 2020 and continue to advocate for women and families.

The DNC raised the bar to get into the third debate, and more than half of the 21 candidates in the race are expected not to qualify.

The candidates now must meet both polling and donor thresholds, instead of either, and each threshold is higher than it had been. Candidates now need 2% in four recognized polls and 130,000 donors across 20 states, up from 1% in polls or 65,000 donors required for the first two debates.

Billionaire Tom Steyer appeared to be on the verge of learning the hard way that money can’t buy everything.

After investing more money than any other individual in Democratic causes over the years, Mr. Steyer had dumped millions more of his largesse into television ads that were supposed to help him snag a debate invite.

But the former hedge fund titan’s hopes took a big hit after a pair of surveys released Wednesday from Suffolk University/USA Today and Quinnipiac University that showed he failed to reach 1%, leaving him a single poll shy of qualifying.

Rep. Tulsi Gabbard of Hawaii and New Age guru Marianne Williamson met the donor thresholds, according to their campaigns, but fell short on polling.

Montana Gov. Steve Bullock, former Rep. John Delaney of Maryland, and New York City Mayor Bill de Blasio also missed the cut.

Mr. Delaney, at least, vowed to march on.

“I’m not going anywhere,” Mr. Delaney said Wednesday on MSNBC, likening the DNC to Thanos, the genocidal villain in the Marvel Cinematic Universe.

“You know, the way the DNC decided to do these debates is they effectively cut out half the field. I don’t think, ultimately, voters want that to happen,” he said. “I think they’re kind of like Thanos, snapping their finger and trying to get rid of half the field.”

Mr. Bullock and Mr. Delaney both did not hit 2% in any approved and also came up short on the donor count.

Mr. Bullock lamented the DNC rules have proven to be a financial boon for tech giants like Facebook and Google.

“Our rules have ended up less inclusive than even the Republicans, which is unfortunate,” Mr. Bullock said on MSNBC. “I think we’ve empowered places like Facebook and Google, but it is what it is.”

Much has been made of the resources campaigns have been spending online to try to raise their national profile and secure the donations and polling support needed to make the debate stage.

DNC Chairman Tom Perez labored to restore confidence in the party’s ability to referee a primary contest after it favored former Secretary of State Hillary Clinton in 2016. He made the process more transparent and used a measure of grass-roots support as well as polling to qualify candidates for the debates.

Suspicions of a rigged 2016 primary were confirmed by DNC officials’ emails, which were hacked — intelligence officials believe by Russian hackers — and then published by WikiLeaks. The emails showed the officials plotting to undermine the candidacy of Mr. Sanders.

Mr. Sanders‘ hardcore supporters remain skeptical of the DNC, despite his being on stage in every debate so far.

DNC spokeswoman Xochitl Hinojosa dismissed the notion that Democratic candidates have been forced to spend money that could have gone elsewhere.

“In order to spend money on the ground, in order to hire organizers, in order to talk to the American public, you need money to pay those organizers,” she said. “You need money to go and crisscross Iowa and New Hampshire and South Carolina and Nevada, and the entire country.”

“So no, we believe empowering the grassroots only helps the Democratic Party and Democratic candidates, and if you have a message that resonates, it shouldn’t be costly,” she said. “If you have a message that resonates, 130,000 [donors] isn’t necessarily a big lift.”

• David Sherfinski contributed to this report.

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