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Tearing up at times, Uber Technologies Inc. Chief Executive Travis Kalanick reflected on the previous five years after a stretch of negative headlines.

“I realize that I can come off as a passionate advocate for Uber,” Mr. Kalanick said, reading from a teleprompter. “I also realize that some have used a different ‘a’ word to describe me. I’ll be the first to admit I’m not perfect, and neither is this company.”

The humbling moment in a June 2015 anniversary speech at Uber headquarters in San Francisco suggested Mr. Kalanick was maturing as a CEO. Nearly two years later, the 40-year-old is again pledging to reform following a spate of setbacks that some investors say has exposed leadership shortcomings at the ride-sharing company, the world’s most valuable startup.

In recent days, people familiar with the matter said, Uber executives have discussed hiring a new senior deputy for Mr. Kalanick, possibly in the mode of Sheryl Sandberg, who as Facebook Inc.’s chief operating officer has served as a public face of the company. The challenge for Uber’s board, composed mainly of its executives and investors, is to make any changes while preserving the formula that built its $68 billion valuation.

Uber declined to comment and said Mr. Kalanick wasn’t available for an interview.

A self-described “hustler,” Mr. Kalanick has a pugnacious leadership style that helped Uber bulldoze into new markets while battling local regulators and competitors around the world. Investors have pumped some $13 billion into the business, which matches some 1.5 million drivers with millions of riders globally, who can hail a car via a smartphone app, often at a lower cost than taxi fares.

Uber’s work culture reflects Mr. Kalanick’s confrontational approach, say employees and investors. The company lists “toe-stepping” and “hustlin’” as corporate values. It has acknowledged aggressive tactics including Operation Slog to divert rival Lyft Inc.’s drivers from picking up legitimate fares, and a program called Greyball to identify and block authorities from using Uber’s app so they can’t interfere with its operations.

The approach has left a swath of detractors, including regulators, rivals, employees and drivers.

Still, Mr. Kalanick, another Uber co-founder and its head of global operations together retain a majority voting stake in the company, and therefore effectively control his fate as CEO, the people familiar with the matter said. Even with a steady drumbeat of negative stories over time, Mr. Kalanick hasn’t faced much internal pressure, being rewarded instead with a skyrocketing valuation particularly as regulatory victories have mounted.

A customer backlash erupted in January when it appeared Uber tried to undermine a brief work stoppage from New York City taxi drivers who were protesting President Donald Trump’s travel ban. Uber said a tweet stating it wasn’t raising fares during the protest was misunderstood. Days later, Mr. Kalanick stepped down from a presidential advisory board.

Last month, a former engineer accused the company of permitting sexism and sexual harassment, prompting others to come forward with similar accusations. Mr. Kalanick condemned the behavior described by the woman and ordered an investigation led by former U.S. Attorney General Eric Holder. In a teary presentation at a town-hall meeting, Mr. Kalanick faced pointed questions from employees upset about what they described as a challenging workplace and tarnished brand.

In late February, Google parent Alphabet Inc. sued Uber, alleging it conspired to steal design plans for self-driving vehicle technology, claims Uber said it was reviewing. Then a dashboard camera video of Mr. Kalanick yelling expletives at an Uber driver emerged, prompting the executive to apologize and pledge to “grow up” and get “leadership help.”

“The question for many entrepreneurs is whether they can shift to a more conventional leadership style, while still maintaining their willingness to defy conventions,” said David Bach, associate dean at Yale University’s School of Management. “The last four weeks or so have unraveled a lot of the work Uber has done.”

At Uber, Mr. Kalanick’s relentless drive is often on display. Tables in some conference rooms are pushed closer to one wall to give him room to pace during meetings. He often spontaneously approaches employees at their desks to ask what they are working on. A little over a year ago, after Mr. Kalanick was seen in the same clothing for several days straight, a few employees surmised he was sleeping at the office and persuaded him to go home and clean up, according to a person familiar with the matter.

Mr. Kalanick has caused, and weathered, previous controversies. He once joked in an interview that the company should be called “Boob-er” because of his ability to attract female companions, and he mocked customers online for complaining about pricing.

Several months before the five-year anniversary speech, a senior company executive was quoted describing plans to investigate journalists who wrote critically about it. The comments, and news that Uber used an internal tool called “God View” that let employees track movements of specific riders, triggered an inquiry from Sen. Al Franken (D., Minn.). Uber scaled back the use of “God View.”

Mr. Kalanick has sought to clamp down on leaks. The release of documents related to internal programs such as Operation Slog prompted Uber to begin to erase all emails after six months. “The answer was not about changing the behavior, it was about changing the disclosure,” a former employee said.

Throughout, investors have largely remained loyal, saying his directness and boldness are rare virtues. And Uber’s rise has created what could be a once-in-a-lifetime payout if the company goes public.

“He definitely has my confidence, he has the board’s confidence,” Arianna Huffington, one of Uber’s only independent directors, told CNBC on Friday. She is assisting the internal investigation into the sexual-harassment claims.

Early investors Freada Kapor Klein and Mitch Kapor have called out in an open letter what they saw as a “toxic” culture at Uber and insufficient responses to prior scandals. But they didn’t suggest change in the c-suite.

“Until there’s a real impact to the fundamentals of the business, say ride requests falling or fewer drivers signing up, investors are likely to view this as just another blip,” said Yale’s Mr. Bach. “Otherwise, Kalanick is going to be their guy.”