Could you add host-side configuration of an upper limit on video length, after which the remainder is automatically skipped? That would handle the case where someone (accidentally or otherwise) chooses an excessively long video (for instance, a version looped for several hours), to avoid the host having to manually skip it.
Related to that, you might consider letting the host designate a privileged client, so that the person running the party can hook that up to their phone to control skips and similar, without going directly to the party machine hooked up to the big screen and sound system. You might also offer options for consensus skips: "if more than N people want to skip a song...".
I like that if you fullscreen the video, it stays that way when a new video comes up; that makes this useful for more than just music. (Please do keep that working if you add non-YouTube support.) However, you might consider making a fullscreenable container that puts the video at almost-fullscreen, but still shows Playlistful and the party ID.
It looks like on the Playlist screen you replace the username with "You", even as a substring of the user's name: I tried it out with users "Foo", "Not Foo", and "Random", and when "Not Foo"'s song came up, Foo's Playlist screen showed it as "Chosen by Not You".
You might consider letting people add a YouTube playlist, or for that matter let users with a YouTube account choose from their favorited/liked/etc videos.