In September 2015, three executives walked into the Tokyo office of Toyota Motor Corp. President Akio Toyoda with a call for a radical change. Toyota, they believed, needed to embrace the goal of building cars that could drive themselves, possibly without drivers—something Mr. Toyoda, a race-car enthusiast who likes his hands and feet on the controls, had long resisted.

The three faced Mr. Toyoda in the office, which one of them recalled felt like a teenage boy’s room, sporting minicars and race helmets. They were ready to...