LOS ANGELES—Netflix added acclaimed filmmaker Steven Spielberg to its roster with a joint announcement on Monday of a deal for his Amblin Partners production company to supply multiple movies a year for several years. The Academy Award-winning director of big-screen classics such as “E.T. the Extra-Terrestrial,” “Schindler’s List” and “Saving Private Ryan” will continue to direct and produce movies for Comcast Corp’s Universal Pictures under a separate agreement. The new arrangement brings a high-profile name to the list of talent working with Netflix at a time when a host of competitors including Walt Disney and Amazon.com are competing for streaming audiences. Spielberg had been at odds with Netflix in recent years when he argued that movies seen primarily on television should be eligible for Emmys and not Oscars. He has spoken out about wanting to preserve the experience of seeing movies in theaters. “My entire life has been spent trying …