Dang it. Please bear with me. My webcam is struggling right now. Sometimes it works, and sometimes it doesn't. I never know until I finish a video whether or not it's going to be all pixellated and gross. This one, unfortunately, is indeed pixellated and gross. I have a new webcam on the way, so hopefully this won't be a problem much longer.
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