LOS ANGELES—When movie executive Scott Stuber left a production deal at Universal Pictures in 2017 to become head of original films for Netflix Inc., he didn’t take a stapler or a box of pens with him. He took “Bird Box.”

Nearly two years later, the horror movie starring Sandra Bullock scored a big audience on Netflix. The film turned into a cultural and social-media phenomenon after premiering on the streaming-video service on Dec. 21.