Long time no see! I'll be a little bit absent for the next few weeks as I have a lot going on but hope to be back as soon as possible. Apologies for the sirens in the background!
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