Courtesy of my friend Dan Peterson, I learned I have been doing Apple TV wrong all these years.
While browsing a list of episodes in a TV show you own, you don’t need to press Select on an episode, then wait for that episode’s individual page and description to load, and then select Play. You can just press the Play button on the episode in the list to start playing it.
If you were in the middle of the episode, you will still get the option to resume or start from the beginning. Also, this annoyingly doesn’t work for movies. While browsing your grid of purchased (and rented) films, pressing the Play button on a film will play its trailer, not the film itself. I hope Apple fixes that in an update soon.