A slight off-the-beaten-path trip to buy some eyeliner resulted in a BUNCH of purchases today...Plus an off-topic ramble about a future YouTube endeavor.
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