Note: I forgot to remove that ring after doing a roleplay. I am NOT engaged! This is a pretty specific request...tapping on glass with long nails, and an inaudible whisper.
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