To the 3.5 people whom noticed, Turns out the balance issue I've been having for a while was in fact a bad microphone cable. I soldered my own right angle connectors previously and must have been a little off. Got some unmodified cables now, so both sides have perfectly matched average rms.
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