FREMONT, Calif.—Elon Musk has always run his car company at his own pace, but now he finds himself in a historic race to deliver on enormous promises. The next six months will determine whether Tesla Inc.’s Model 3 can reshape the U.S. auto industry or serve merely as a flashy footnote.

Tesla’s race is unfolding on the floor of a factory Mr. Musk acquired here in Northern California from Toyota Motor Corp. earlier this decade. I took a tour Thursday, observing the body shop where 1,068 robots were welding, gluing or riveting...