I made a playlist of the songs in this video here: https://www.youtube.com/watch?v=ZhALJhMMYkk&index=1&list=PL6LjLUQJEXroiVErZAjLAh8L21NskgGfU
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