For anyone else who hasn't come across it: Newpipe allows you to (a) watch the video directly in fullscreen (b) download it as video or audio (c) play the audio in the background (d) play it in a small (adjustable) pop-up window that can remain on screen while you browse other apps.

It's still very much in development though. There are random errors that pop up when navigating (not too frequent, but you'll probably see one if you use it for a week or so), and it just recently got the ability to display playlists (you have to manually select the next video after one finishes playing, doesn't have auto-advance feature yet).