Pardon my rant here. I've got a bee in my bonnet about the current goings-on in my life, and I needed to vent a little bit. Making that roleplay about Lynette's situation must've cursed me, because now I'm going through the same thing. ☹️
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