Several people asked me to do another video of this type, so here we have it. This video features, I believe, several flavors that we didn't have in the first two. (I could be wrong, but it's late and I'm too lazy to go back and check.)
To listen while using other apps, switch the player to Picture-in-Picture (PiP) during playback — it keeps playing in a small floating window (the screen stays on).
To listen with the screen fully off, in-browser playback stops by YouTube's design. Open the video in the YouTube app to keep listening where background playback is supported (e.g. with YouTube Premium).
Open in YouTube app