Just screwing up but had to share how much of the intro to Let’s Go Crazy is swimming around in my mind! You can take the girl out of the 80's, but you can't take the 80's out of the girl!
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