I'm doing any% runs and as great as the soundtrack is, I'm starting to get a little tired of it. Wanted to clarify if the recommendation is "game sounds[,] and music" or "game sounds and [game] music", and if people would rather no music but game SFX or game SFX + own music

in-game music is not ever required but in-game sound is recommended (and required for top level runs)

Just to clarify - game sounds/music usually come into play for:
- Runs that are particularly noteworthy, usually meaning a top-level/podium time. Even then we'd usually try to get extra proof of legitimacy for these runs that pose an issue in meeting our verification standards rather than immediately rejecting.
- Runs with a significant portion of the footage lost to frame drops. We'd usually try to rely on game audio of some kind to ensure that the game is actually still being played during those frame drops, so if it's not present then it becomes increasingly difficult to justify verification.

The recommendation should be read as "game sounds and [game] music", so whether you choose to include your own music or not in the recording is totally up to you - if game sounds are still audible then either option shouldn't be a problem for verification.
If you want a personal view on it, I pretty much exclusively play with no game music and my own in the background, but I make sure it's not recorded to avoid copyright strikes on VODs and all that. Might make for a better viewing experience if you have killer taste though :​)