WASHINGTON — Kamala Harris was frustrated. The text of a speech she had been given to deliver in Chicago to the nation’s biggest teachers’ union was just another dreary, scripted talk that said little of any consequence.As Air Force Two made its way to the Midwest over the summer, the vice president told her staff she wanted to say something more significant, more direct. She brandished a Rolling Stone magazine article about the backlash against Florida school officials after new legislation barring the discussion of gender identity in the classroom.The teachers she was about to address were on the front lines of the nation’s culture wars, Ms. Harris told her staff. They were the same ones on the front lines of school shootings. Just blandly ticking through federal funding for education would not be enough. The plane was just an hour out from Chicago, but she said they needed to start over.By the time she landed, she had a more spirited version of the speech in hand, accusing “extremist so-called leaders” in the Republican Party of taking away rights and freedoms.Ms. Harris’s small airborne rebellion that day encapsulated the trap that she finds herself in. She has already made history as the first woman, the first African American and the first Asian American ever to serve as vice president, but she has still struggled to define her role much beyond that legacy.Her staff notes that she has made strides, emerging as a strong voice in the administration on abortion rights. She has positioned herself as a more visible advocate for the administration, giving a speech last week at the funeral for Tyre Nichols, the 29-year-old who was beaten by Memphis police officers. And her critics and detractors alike acknowledge that the vice presidency is intended to be a supporting role, and many of her predecessors have labored to make themselves relevant, as well.But the painful reality for Ms. Harris is that in private conversations over the last few months, dozens of Democrats in the White House, on Capitol Hill and around the nation — including some who helped put her on the party’s 2020 ticket — said she had not risen to the challenge of proving herself as a future leader of the party, much less the country. Even some Democrats whom her own advisers referred reporters to for supportive quotes confided privately that they had lost hope in her.Through much of the fall, a quiet panic set in among key Democrats about what would happen if President Biden opted not to run for a second term. Most Democrats interviewed, who insisted on anonymity to avoid alienating the White House, said flatly that they did not think Ms. Harris could win the presidency in 2024. Some said the party’s biggest challenge would be finding a way to sideline her without inflaming key Democratic constituencies that would take offense.Now with Mr. Biden appearing all but certain to run again, the concern over Ms. Harris has shifted to whether she will be a political liability for the ticket. Given that Mr. Biden at 80 is already the oldest president in American history, Republicans would most likely make Ms. Harris, who is 58, a prime attack line, arguing that a vote for Mr. Biden may in fact be a vote to put her in the Oval Office.The Run-Up to the 2024 ElectionThe jockeying for the next presidential race is already underway.“That will be in my opinion one of the most hard-hitting arguments against Biden,” said John Morgan, a prominent fund-raiser for Democrats, including Mr. Biden, and a former Florida finance chairman for President Bill Clinton. “It doesn’t take a genius to say, ‘Look, with his age, we have to really think about this.’”So far, he said, she has not distinguished herself. “I can’t think of one thing she’s done except stay out of the way and stand beside him at certain ceremonies,” he said.Some 39 percent of Americans approve of Ms. Harris’s job performance, according to a recent aggregate of surveys compiled by the polling site FiveThirtyEight. This puts her below Mr. Biden’s approval rating, which has hovered around 42 percent for the past month.Ms. Harris’s allies said she was trapped in a damned-if-she-does, damned-if-she-doesn’t conundrum — she is expected to not do anything to overshadow Mr. Biden while navigating intractable issues he has assigned her such as voting rights and illegal immigration. And some see a double standard applied to a prominent woman of color.“That’s what being a first is all about,” said Representative James E. Clyburn, Democrat of South Carolina and one of the nation’s most prominent Black lawmakers, who has been an outspoken supporter. “She’s got to work every day to make sure she’s not the last.”While Mr. Biden was quoted in a new book by Chris Whipple, “The Fight of His Life,” calling Ms. Harris a “work in progress,” the White House defended her when asked for comment, forwarding a statement from Ron Klain, the president’s departing chief of staff who has been her most important internal ally.Mr. Klain, who served as chief of staff to two vice presidents, said that those who hold that post often “take grief” but go on “to prove skeptics wrong.” He cited Ms. Harris’s outspoken support for abortion rights and her international trips. “She has done all that operating under high expectations,” he added, noting her status as various firsts. “She carries these expectations not as a burden but with grace and an understanding of how much her history-making role inspires others.”Ms. Harris has a fresh opportunity to find her footing with the arrival of the new Congress. Because the Senate was split evenly for the last two years, Ms. Harris has cast 26 tiebreaking votes in her role as president of the Senate, more than any vice president since John C. Calhoun, who left office in 1832. Tethered to Washington, she could never be more than 24 hours away from the Capitol when the Senate was in session in case her vote was needed.With Democrats now holding a 51-to-49 edge, at least in cases when Senator Kyrsten Sinema of Arizona, the rogue Democrat-turned-independent, votes with them, Ms. Harris has a little more breathing space. She has told her staff that she wants to make at least three out-of-town trips a week in the coming year.No one feels the frustration of being underestimated more acutely than Ms. Harris, but she makes a point of not exhibiting it publicly. In an interview with The New York Times while she was in Japan last fall, she tried to explain her own political identity.What we consider before using anonymous sources. Do the sources know the information? What’s their motivation for telling us? Have they proved reliable in the past? Can we corroborate the information? Even with these questions satisfied, The Times uses anonymous sources as a last resort. The reporter and at least one editor know the identity of the source.“You got to know what you stand for and, when you know what you stand for, you know what to fight for,” Ms. Harris said.What that translates to in tangible terms is less clear. After her disastrous interview with Lester Holt of NBC News in June 2021, in which she struggled to articulate the administration’s strategy for securing the border, White House officials — including some in her own office — noted that she all but went into a bunker for about a year, avoiding many interviews out of what aides said was a fear of making mistakes and disappointing Mr. Biden.Members of Congress, Democratic strategists and other major party figures all said she had not made herself into a formidable leader. Two Democrats recalled private conversations in which former Secretary of State Hillary Clinton lamented that Ms. Harris could not win because she does not have the political instincts to clear a primary field. Nick Merrill, a spokesman for Mrs. Clinton, said she was strongly supportive of Ms. Harris and often spoke with her about shared experiences of being “a woman in power.” He added: “They have built and maintained a strong bond. Any other characterization is patently false.”Advisers and allies trace Ms. Harris’s challenges to her transition from the lawyerly prosecutor she used to be as district attorney of San Francisco and attorney general of California into a job where symbolism and politics are prioritized.Aides have encouraged her to liberate herself from the teleprompter and show the nation the Ms. Harris they say they see when the cameras are off, one who can cross-examine policymakers on the intricacies of legislative proposals and connect with younger voters across the country.Ms. Harris has acknowledged her reservations about leaning into the more symbolic aspects of her current position.“My bias has always been to speak factually, to speak accurately, to speak precisely about issues and matters that have potentially great consequence,” she said in the interview in Japan. “I find it off-putting to just engage in platitudes. I much prefer to deconstruct an issue and speak of it in a way that hopefully elevates public discourse and educates the public.”Ms. Harris finds herself navigating the unique dynamics of being a woman of color in a job previously filled only by men. In planning meetings before she travels abroad, officials from foreign governments have proposed meetings or public appearances with the first lady of the country Ms. Harris is visiting. Her staff rebuffs those proposals, saying the vice president is not visiting as a spouse but as the second-ranking official of the United States, according to current and former White House officials. There are more mundane hiccups, as well. Jamal Simmons, who recently stepped down as communications director for the vice president, said he learned that the desk chairs in her office needed to be changed to suit Ms. Harris — who stands about 5-foot-2 — instead of the “average male height” of her predecessors. “She forces us to recalibrate our assumptions,” Mr. Simmons said. Ms. Harris has, at times, expressed hesitation to become the face of certain issues. When the Biden administration confronted a shortage of baby formula across the nation last year, Ms. Harris declined a request by the West Wing to highlight efforts to solve the problem by meeting a shipment of formula at Washington Dulles International Airport, one current and two former administration officials said, speaking on the condition of anonymity to describe the decision. Instead, Jill Biden, the first lady, ended up appearing alongside the surgeon general when the shipment arrived from overseas. (Nearly a month later, Ms. Harris did agree to meet one of the shipments.) Ms. Harris disputes the idea that she is concerned about being assigned — or pursuing — certain tasks solely because of her gender or identity.“I’m fully aware of stereotypes, but I will tell you something: I’ve never been burdened by a sense of ‘I should not do something that’s important because I will be pigeonholed,’” Ms. Harris said during the interview in Japan. She said she had pursued the abortion rights issue, for example, “because I feel it is one of the biggest tragedies that has happened at this level of our government in a very long time.”Ms. Harris often tells senior aides that she feels most comfortable receiving intelligence briefings or addressing law enforcement officials, venues where she says substance is valued over politics. She has directed staff members to ensure that she is making trips to speak about the administration’s accomplishments, such as the Inflation Reduction Act, and not just the multiple crises it faces. She has also peppered her staff with questions about local abortion access and how the decision overturning Roe v. Wade could lead to criminalization of medical officials.“She has her prosecutor hat on that way,” said Alexis McGill Johnson, the president of Planned Parenthood, who has watched the vice president try to distill complex health care issues in a way that “everyday citizens” can understand.And months after she revised her Chicago speech aboard Air Force Two, Ms. Harris went through nine drafts before delivering a speech in Tallahassee, Fla., on the 50th anniversary of Roe, in which she asked if Americans can ever “truly be free” if a woman cannot make decisions about her own body.Several attendees said they were encouraged to see a Black woman speaking clearly about how threats to Roe represent a broader threat to civil rights.It was “very powerful for me to see someone with my likeness in this position in this day and age,” said Sabrita Thurman, 56, who is Black.Those close to Ms. Harris hope she can move beyond “defensive politics,” said Douglas Brinkley, a presidential historian who organized a meeting at her residence about the legacy of the vice presidency and will attend another session with her this week.“President Biden has to give her more leeway to be herself and not make her overly cautious that a mistake, a rhetorical mistake, will cost the party a lot,” Mr. Brinkley said. “It’s better to let Kamala be Kamala.”Michael D. Shear contributed reporting, and Kitty Bennett contributed research.