Mr. Lasseter’s high opinion of Mr. Staggs may seem beside the point, but that kind of warm connection is critical to Disney’s future. In many ways, the company’s continued success turns on the ability of Mr. Staggs — should he indeed win Mr. Iger’s crown — to sustain a corporate environment where creative powerhouses like Mr. Lasseter are happy and can flourish.

Mr. Iger’s tenure has largely been about acquiring boutique content factories, including Lucasfilm, the “Star Wars” studio purchased in 2012 for $4 billion. To continue to grow, Disney needs those creative assembly lines to keep humming, a difficult task given the imagination-numbing realities of life inside a mega-corporation.

The theme park division that Mr. Staggs led is much bigger than most people realize. Walt Disney Parks and Resorts employs 130,000 people on three continents, with 13 parks attracting an estimated 133 million visitors annually. Rounding out the division: four Disney Cruise ocean liners, about 40 hotels, a condo time-share empire, the stand-alone Aulani resort in Hawaii and the Adventures by Disney private tour business.

The challenges come in waves: The violent tossing of the 4,000-passenger Disney Fantasy cruise ship in a hurricane. A helium shortage threatening balloon sales. Construction hiccups at Shanghai Disneyland. The unveiling of $1 billion in improvements at Disneyland Resort in California. Mr. Staggs once had to contend with all of those in the span of six months.

Mr. Staggs is also credited with sorting out technological problems with MyMagic Plus, a new Walt Disney World visitor management system that changes how Disney World visitors do just about everything. The $1 billion system, designed to increase the length of time people stay at the sprawling resort, involves equipping guests with data-collecting electronic wristbands. Started under Mr. Rasulo, who declined to comment for this article, the system had a difficult birth because of technology hurdles and infighting. Mr. Staggs brought in new managers and pushed them to look harder at emerging technology. The system is now running smoothly.

Still Much to Prove

Mr. Staggs’s succession is not a given. Disney’s board has made it clear that he must still prove he has what it takes to be chief executive. Part of that insistence may reflect Disney’s history in choosing leaders. The bumpy transition of power from Mr. Eisner to Mr. Iger in 2005 became an example of what not to do. It got so bad that Roy E. Disney, the nephew of Walt Disney, used the lack of succession planning in a public attack on Mr. Eisner.

And, truth be told, Mr. Staggs has some dents in his armor.

He has not made a firm ally of Isaac Perlmutter, the strong-willed Marvel chief executive, who has been a vocal supporter of Mr. Rasulo.