Tesla has reportedly been searching for years for a No. 2 to ease CEO Elon Musk's workload, even approaching Facebook Chief Operating Officer Sheryl Sandberg at one point. Executive recruiters and industry analysts agree Musk needs help.

Musk's erratic behavior on Twitter has reportedly drawn a federal securities investigation and put Musk himself at risk of criminal charges, securities lawyers say. He drew criticism for calling a diver helping to rescue boys trapped in a cave in Thailand a "pedo." His mental fitness was also called into question in a lengthy interview he gave the New York Times two weeks ago complaining about the stress of trying to keep up with demand for the Model 3 sedan.

"It is time for Tesla's board to consider that as a publicly held company it has a responsibility to the shareholders and to the public," said Dale Jones, president of executive recruiting firm Diversified Search. "When you move from a founder led organization, you have to move to more of a team-led organization where you have leaders that are complemented by other leaders who bring strengths you don't have."

Tesla didn't directly address the search for another executive to help Musk in a statement from the board of directors released to CNBC.

"There have been many false and irresponsible rumors in the press about the discussions of the Tesla board," the board said. "We would like to make clear that Elon's commitment and dedication to Tesla is obvious. Over the past 15 years, Elon's leadership of the Tesla team has caused Tesla to grow from a small startup to having hundreds of thousands of cars on the road that customers love, employing tens of thousands of people around the world, and creating significant shareholder value in the process."

Still, recruiters and industry watchers say the company may need to add more than one person to its executive ranks to help keep Musk on task and from tweeting himself into more hot water.

Although Musk told the New York Times Tesla wasn't able to get Sandberg, Jeff Cohn, managing partner at leadership advisory firm Elevate Partners, said a woman would be a good fit for Musk.

"Someone who has those Sheryl Sandberg-esque qualities that could figure out how to develop the kind of relationship with Elon that is candid and transparent, so she could be a coach to him," Cohn said. "These soft skills for a CEO I think would be at least as important as the harder skills of knowing logistics, supply, and global operations."

He said women have an innate advantage over men when it comes to empathy and emotional intelligence. "That would really play well when you are dealing with a strong ego, like Elon," he said.

Emotional intelligence aside, Musk currently has a more practical and pressing problems at his plants. The automaker has repeatedly missed production targets, has rarely been able to earn a profit and has been criticized for quality issues with its vehicles and problems getting timely service for customers.

Some of this results directly Musk's tendency to make big promises and take on too much, said Karl Brauer, executive publisher for Cox Automotive which provides a range of services to the auto industry.

"The smartest thing that Elon could have done was hired John Krafchik after he left Hyundai in 2013," Abuelsamid said. Krafchik, who was instead later scooped up to run Google's self driving project Waymo, is a longtime automotive industry executive who understands many of the complex aspects of running a major automaker, including manufacturing, engineering, labor, and relationships with suppliers.

Brauer also suggested Aston Martin CEO Andy Palmer. Aston Martin is the strongest it has been in its 100 year history, he said. Palmer has a lot to do with that. Palmer has deep experience in the industry. He worked for Nissan, a high-volume manufacturer, and understands how to make cars.

"He and brought that grounded industrial perspective to a Aston Martin," Brauer said.

Someone like John Martin at Nissan North America also comes to mind, said Dave Sullivan, an analyst for industry research firm AutoPacific. Martin is senior vice president of manufacturing, supply chain management and purchasing at Nissan. He is known as a financially savvy, no-nonsense executive and has experience at a large industrial automaker.

John Savona, who is director of manufacturing for assembly at Ford could also bring some rigor to Tesla's production issues. Savona is a quietly rising star at Ford. Sullivan said he has an impressive ability to handle manufacturing problems.

Tesla may have trouble recruiting because a lot of industry veterans at large manufacturers might not want to give up a stable job to leave for a company in as much turmoil as Tesla. And they might not want to work for a boss who is reported to control everything down to the color of paint on the factory floor.

"The problem is, the house is on fire," Sullivan said. "No one really wants to run in. You might not make it out alive."

Musk is largely in this position for several reasons, say industry analysts. Trying to build an automotive manufacturer in this competitive market is difficult enough. But Tesla is extremely unusual. For example, it owns a solar power and energy storage business and owns its own stores and service centers, something conventional automakers don't do. Many have also criticized Musk for setting unrealistic goals and targets, and for failing to learn from the mistakes and moves made by other automakers over the last century.

For example, Musk said he wanted to hit a Model 3 production rate of 5,000 vehicles a week by the end of 2017. Many said the goal was unrealistic. Tesla didn't hit that goal until the end of June 2018. Likewise, the Model 3's predecessor, the Model X SUV, was delayed largely due to all of the complex and high-tech features Tesla stuffed into the car.

"The rule in business, perhaps in everything, is you under-promise and you over-deliver," Brauer said. "And I don't think it is inaccurate to say Tesla has fairly consistently done the opposite. That is the nexus of what restricts Tesla from being what it could be."

While Musk has said he wants to revolutionize automotive manufacturing, Tesla has repeated mistakes made by other automakers decades ago.

For example, Musk was criticized for relying too heavily on automation at its Fremont factory, especially in the later stages of the assembly process. It has long been believed some processes are simply too difficult to automate. Musk later said as much. "It tuns out humans are underrated," he said.

Humans are underrated

But General Motors had made that mistake in the 1980's when then-chairman and CEO Roger Smith was a serious proponent of automation, to the point of believing that GM could have factories so automated they could run in the dark, said Sam Abuelsamid, a senior research analyst for industry researcher Navigant Research. It was a disaster for GM.

These are the kinds of errors the company can avoid with input from industry veterans.

"I think one of the issues with Elon and maybe a lot of smart people is overconfidence," he said. "It is important to have visionary ideas, but you also need to have some humility."