I couldn't listen to music while playing Red Dead. The music in-game is incredible and I would never turn it off for something else.
That being said, Faith No More's cover of the Midnight Cowboy theme song would be perfect for a ride sequence in RDR1 where there's music playing in the background (so like Far Away when you're in Mexico for the first time, or Compass when you're riding home to your family). I'm just not sure at what point should it have played.