I'm not trying to attack you personally - just discussing (maybe pontificating?), perhaps too emphatically, an interesting and important topic of game design: Are stories important in games?
Am I straying too far off the original topic? (If so, just let me know - I'm not trying to hijack the thread. Perhaps I misunderstood what the original topic was even asking about)
Plus, music and sound are totally different. Sound is part of how you experience the world - the real world. Walking around outside what you see and hear are nearly as important as each other. But I rarely get background music in real life 
My point is that music is also part of how you experience the game as a whole, and story is also part how you experience the game as whole. Neither music, sound, or story should exists in isolation separated from the rest of the game.
If a specific story (or specific background music) really doesn't benefit a specific game, either the story was badly done, or that specific game probably is better off without a story. But story, in general and when done right, is important in games and can't just be removed, muted, or ignored without damaging (even if just minorly) the total experience of the game.
Thankfully, two days ago this video came out (after I originally posted, so I wasn't influenced by the video - it just put my existing thoughts into a different coherent form and looked at them from a different angle), which explains a similar thing from a different perspective. 
Leastwise, that's my opinion.
Halo (or Chrono Trigger, or Morrowind, to use non-FPSes), without it's music, wouldn't be as Halo-y as it is with its music. If I muted Halo's soundtrack, I'm muting more than just the music, I'm also muting part of the game that is meant to go together with the scripted events and action and level design to compliment them and to set a deeper-level emotional reaction that is then used by the rest of the game to augment the overall experience.
If I muted the music and played, say, violin music, it'd still be really enjoyable, but it wouldn't be as fitting as the music wouldn't fully partner with the level and the action. It'd just co-exist alongside in mutual ambivalence to each other. Now, if someone carefully created an alternative soundtrack that fit Halo and that was intentionally designed to augment and amplify the Halo experience, that's a different matter! But if I just run whatever playlist I'm listening to, it won't be as fitting, and sometimes may even be detrimental (playing an exciting upbeat song during what's supposed to be a sad moment, or similar conflicts).
If I ignored the plot of Halo, removed all the cutscenes, and basically just relegated it to Doom (as far as story goes), it'd still be an enjoyable game... but (I theorize, never having actually done this) not as enjoyable as when the Halo story, if written well, compliments your in-game goals and your in-game character, and colors the enemies you face, the world you explore, and the events you encounter.
I've certainly played many games where the story was 'meh', and I ignored it. Sometimes the story was 'good', but didn't fully compliment the rest of the game (just layered over it instead of woven through it), so I tuned it out. I think that is a failing of the developers, not an indication that story, as one of the most important game-design tools, is irrelevant or unimportant in games.
You said, "I personally don't really care [about story in games], whether it's an FPS or RTS I just want to play the level rather than go through some (usually trite) backstory."
The part I highlighted is where I think the problem is: Not that story is unimportant, but that you are usually running into poorly written stories slapped over almost as an afterthought or (more likely) the story was gutted and chopped up (or stretched out and watered down) as the nature of the game changed during development.
This is not a matter of games and stories not going together, but that most games just have bad stories or poorly implemented stories - just like for a long time, most 3rd person games had bad camera angles. That doesn't mean 3rd person doesn't work in games, it just meant we needed to invest more effort into handling cameras better in games (which we did successfully and now have fewer games ruined by bad cameras).
I fully agree that not all games should have a story crammed into it. I also might agree (I would need to put some more thought into it) that not all players enjoy stories, even if the story was done perfectly and implemented perfectly and perfectly complimented the rest of the game. And I point back to my hastily slapped together chart in my first post.
