Millions of people tuned into live broadcasts of the electronic entertainment expo (E3) this week on Alphabet Inc.’s YouTube and Amazon.com Inc.’s Twitch.tv. It’s an event whose viewership has soared on the heels of a phenomenon that has been building for years, spawning a lucrative business that’s attracting competitors, inducing acquisitions and could be affecting traditional television. Hundreds of millions of viewers now watch video of both amateurs and professionals playing various games. Video-game live streaming is an industry that “has truly...