My webcam is gasping for air and on its last legs (hence the recent glitchy, pixellated videos). I'm awaiting the delivery of my new webcam, and while a vlog may not be the best substitute, I'm afraid it's the best I can do at this point...
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