-Software updates.
This is likely the most useful thing it could do.

-Competition.
A global leaderboard would be neat, till someone takes the glass off.

-Bug reports.
Mixed emotions. Seems code has been buggy out of the gate, not sure if this would remedy or exacerbate the situation.

-Playing together.
Forced multiplayer is a terrible idea. Locking modes behind multiplayer only is just dumb.

-Remote access/Service App
This seems like its lumped in with the software updates, and your idea of it sounds like a security nightmare. Unless it was some sort of read only app and the changes had to be done on the actual machine.

-Bluetooth/Player App
Sounds like a solution to a problem that doesnt exist. I personally enjoy putting my initials in the game and Im more likely to put them in using the flippers than I am downloading an app and setting up a profile.