Okay, full disclosure, I most definitely knocked my phone off the hanger at the end... I felt the rest of it was pretty good though and decided to share it anyways. It's a little short because it got ended early, but try and relax for the few minutes it lasts.
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