Am I missing something by skipping the end of episode screen?
You're not missing out on any story relevant content as far as I know. I've stared at the screen when I finished Episode 1 only to realize that the only thing that plays during these screens are songs.
When the song ends, the Episode Ending texts fades into some kind of mist. The song doesn't start again. Instead, the next episode starts (beginning with the recap of the previous episode).