Few, if any, literary philosophers have had as much influence on American business and politics as Ayn Rand, especially now that Donald J. Trump occupies the White House.

President Trump named Rand his favorite writer and “The Fountainhead” his favorite novel. Secretary of State Rex W. Tillerson has cited “Atlas Shrugged” as a favorite work, and the C.I.A. director, Mike Pompeo, said the book “really had an impact on me.”

As Ray Dalio, the founder of Bridgewater Associates, the world’s largest hedge fund, put it in a recent essay, “her books pretty well capture the mind-set” of the Trump administration. “This new administration hates weak, unproductive, socialist people and policies,” he wrote, “and it admires strong, can-do profit makers.”

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In business, Rand’s influence has been especially pronounced in Silicon Valley, where her overarching philosophy that “man exists for his own sake, that the pursuit of his own happiness is his highest moral purpose, that he must not sacrifice himself to others, nor sacrifice others to himself,” as she described it in a 1964 Playboy interview, has an obvious appeal for self-made entrepreneurs. Last year Vanity Fair anointed her the most influential figure in the technology industry, surpassing Steve Jobs.

But lately, many Rand devotees have been running into trouble. Travis Kalanick’s abrupt departure as chief executive of Uber, the Internet-based ride-hailing service he built into a private corporation worth $50 billion or more, is the latest Icarus-like plunge of a prominent executive identified with Rand.

The hedge fund manager Edward S. Lampert, who some say has applied Rand’s Objectivist principles to the management of Sears and Kmart, has driven those venerable retailers close to bankruptcy.

Andrew F. Puzder, Mr. Trump’s first nominee for secretary of labor, is described by friends as an avid Ayn Rand reader. He’s also chief executive of CKE Restaurants, which runs the Hardee’s and Carl’s Jr. fast-food chains and whose private equity owner, Roark Capital Group, is named for the architect-hero of “The Fountainhead.” Mr. Puzder had to withdraw his nomination after allegations that his restaurant companies mistreated workers and promulgated sexist advertising.

The Whole Foods founder and chief executive John Mackey, an ardent libertarian and admirer of Rand, last month had to cede control of the troubled upscale grocery company to Amazon and Jeff Bezos (who, while often likened to a fictional Rand hero, has not mentioned her books when asked about his favorites).

And then there’s the scandal-engulfed Trump administration, where devotion to Rand’s teaching has done little to advance the president’s legislative agenda.

Though people close to Mr. Kalanick told me this week that he has distanced himself from many of Rand’s precepts while undergoing an intense period of personal reassessment, they all acknowledged that she’d had a profound influence on his development. Few companies have been as closely identified with Rand’s philosophy as Uber.

Uber disrupted a complacent, highly regulated and often corrupt taxi industry on a global scale, an achievement Rand’s heroes Howard Roark and Dagny Taggart would surely have admired. Many of her ideas were embedded in Uber’s code of values. Mr. Kalanick used the original cover art for “The Fountainhead” as his Twitter avatar until 2013 (when he exchanged it for an image of Alexander Hamilton, and then, in May, for one of himself).

But Mr. Kalanick was urged to step down as chief executive by the Uber board and Uber’s major investors over less heroic issues: that Uber fostered a workplace culture that tolerated sexual harassment and discrimination; that it ignored legal constraints, poaching intellectual property from Google’s self-driving car endeavor and using technology to evade law enforcement; and that it failed to hire a chief operating officer or build an effective management team. (Mr. Kalanick remains on the board.)

“Rand’s entrepreneur is the Promethean hero of capitalism,” said Lawrence E. Cahoone, professor of philosophy at the College of the Holy Cross, whose lecture on Rand is part of his Great Courses series, “The Modern Political Tradition.” “But she never really explores how a dynamic entrepreneur actually runs a business.”

“She was a script and fiction writer,” he continued. “She was motivated by an intense hatred of communism, and she put those things together very effectively. She can be very inspirational, especially to entrepreneurs. But she was by no means an economist. I don’t think her work can be used as a business manual.”

Representatives of Uber and Mr. Kalanick declined to comment.

Rand’s defenders insist that the problems for Mr. Kalanick and others influenced by Rand aren’t that they embraced her philosophy, but rather that they didn’t go far enough.

Yaron Brook, executive chairman of the Ayn Rand Institute and a former finance professor at Santa Clara University, who teaches seminars on business leadership and ethics from an Objectivist perspective, said, “Few business people have actually read her essays and philosophy and studied her in depth.” Mr. Brook said that while Mr. Kalanick “was obviously talented and energetic and a visionary, he took superficial inspiration from her ideas and used her philosophy to justify his obnoxiousness.”

He emphasized that Rand would never have tolerated sexual harassment or any kind of mistreatment of employees. Rand “had enormous respect for people who worked hard and did a good job, whether a secretary or a railroad worker,” he said. “Her heroes ran businesses with employees who were very loyal because they were treated fairly. Of course, some people had to be fired. But she makes a big deal out of the virtue of justice, which applies in business as well as politics.”

And even though “she’d celebrate what Travis did with the taxi industry, showing the world how all those regulations made no sense, she also believed there are rules of justice that do make sense and she supported,” he said. “You can’t just run over all the regulations you don’t happen to like.”

Mr. Brook complained that Rand’s critics are quick to point to her followers’ failures, but rarely mention their successes. He cited the example of John A. Allison IV, the much-admired former head of BB&T Corporation, a regional bank in the Southeast that he built into one of the nation’s largest before he stepped down in 2008. Mr. Allison handed out copies of “Atlas Shrugged” to senior executives and is a major donor to the Ayn Rand Institute. He incorporated many of Rand’s teachings into his 2014 book, “The Leadership Crisis and the Free Market Cure.”

“John is a gentleman and he actually studied Rand’s works in depth,” Mr. Brook said. “He couldn’t be more different from Travis.”

Mr. Allison has called for abolishing the Federal Reserve, while acknowledging that so drastic a step is unlikely. He has met with Mr. Trump at the White House and has been widely mentioned as a potential successor to Janet L. Yellen as Fed chief.

Despite Rand’s pervasive influence and continuing popularity on college campuses, relatively few people embrace her version of extreme libertarianism. Former President Barack Obama, in a 2012 Rolling Stone interview, criticized her “narrow vision” and described her work “as one of those things that a lot of us, when we were 17 or 18 and feeling misunderstood, we’d pick up.”

She’s also dismissed by most serious academics. “Mention Ayn Rand to a group of academic philosophers and you’ll get laughed out of the room,” Mr. Cahoone said. “But I think there’s something to be said for Rand. She takes Nietzschean individualism to an extreme, but she’s undeniably inspirational.”

As the mysterious character John Galt proclaims near the end of “Atlas Shrugged”: “Do not let your fire go out, spark by irreplaceable spark, in the hopeless swamps of the approximate, the not-quite, the not-yet, the not-at-all. Do not let the hero in your soul perish, in lonely frustration for the life you deserved, but have never been able to reach. Check your road and the nature of your battle. The world you desired can be won, it exists, it is real, it is possible, it’s yours.”

But Rand has little to say about making the transition from this kind of heroic entrepreneurial vision to a mature corporation with many stakeholders, a problem many company founders have confronted and struggled with, whether or not they’ve read or been influenced by her. “She never really had to manage anything,” Mr. Cahoone said. “She was surrounded by people who saw her as a cult figure. She didn’t have employees, she had worshipers.”

For his part, Mr. Kalanick is said to have turned this summer from Rand to what is considered one of the greatest dramatic works in the English language, Shakespeare’s “Henry V” — a play in which the young, reckless and wayward Prince Hal matures into one of England’s most revered and beloved monarchs.