Sorry I've been gone for awhile, guys. Been sucked into this game hardcore. (I know I'll get random people coming to this who don't know what ASMR is and think it's "creepy" to be whispering. Please look up what it is before you comment.)
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