At long last, I have finally managed to get through the whole piece without any major distractions or mistakes. I do apologize for the noise of the keys and the pedal, as I have used my keyboard instead of my piano because it's significantly out of tune at the moment.
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