(Mostly) scratching on a microfiber office chair. This one's a bit silly :P Also sorry about the cable hanging in front later on, not sure how I didn't notice that!
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