Master, you do indeed have some questionable interests (On top of the usual issues of the frog in my throat, the image decided to be blurry =w= I'm sorry, and will try to have the issue solved for next week! (To help my dog Holo please donate to my ko-fi, you can also get an audio commission from doing so.
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