LOS ANGELES — He had a toe in the water.

In 2015, with Netflix growing at a scorching rate, Robert A. Iger, Disney’s chief executive, quietly tested a streaming app in Britain called DisneyLife. The next year he started talking more openly about building a major streaming platform — a risky proposition for a company with a vast traditional TV business — and paid $1 billion for a stake in BamTech, a little-known tech company that had helped HBO build its app.

But it was not until 2017 that Mr. Iger decided to pinch his nose and step off the diving board.

For two days that June, during a board of directors retreat at the company’s luxury BoardWalk Inn at Walt Disney World, Disney executives gave presentations on how digital technology was disrupting their divisions. The most alarming report came from Media Networks, a $24 billion unit anchored by ESPN and the Disney Channel. Cord-cutting was accelerating at a much faster rate than anticipated. Live viewing for children’s programming was in a free fall.

Less than two months later, Mr. Iger announced a radical plan: The Walt Disney Company would shift its priority to streaming by creating ESPN Plus, a platform for sports, and Disney Plus, which would include blockbusters from the Marvel, Pixar and “Star Wars” universes, as well as decades of Disney classics.

“We were now hastening the disruption of our own business,” Mr. Iger wrote of the move in his recently published memoir, “The Ride of a Lifetime.” The chapter title: “If You Don’t Innovate, You Die.”