Our Sony TV has a "Screen off" mode under quick-settings menu. Everything goes off related to screen, incl LEDs. We use it a lot while listening to music via Spotify and other services.
The beauty is, volume buttons don't turn the screen back on either, so it truly becomes a connected speaker.
My old one had a remote button for “PIC OFF”! What year roughly was yours made? I’m wondering if my new one has that buried somewhere. Mine is running the “Google TV” OS
Mine is a Sony X900H[0] (or its equivalent in EU), and runs Android, too. The feature is under the (Sony specific) quick settings menu accessed by the gear icon on the remote control.
The beauty is, volume buttons don't turn the screen back on either, so it truly becomes a connected speaker.