Quick-tip: I’ve been struggling with a second-generation Apple TV for several years, and in the last couple of years, it’s taken longer to start playing a movie or show once I’ve hit “play”. The last couple of weeks it’s gotten to the point where it basically doesn’t start playing at all.
I found a fix: go into Settings > General, and turn off “Send data to Apple”. Power-cycle the Apple TV. This fixes the streaming.