Let me know what you think and how to improve. Sorry about filming it vertically, it was due to how I had to position the ipad to stay up. However that doesn't matter, it's really the sound you're after, right?
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