For games like Mario Kart for example, if Nintendo receives overwhelming feedback on aspects people really dislike (say no option for removing timed battle modes) is there a possibility of being able to update the game via download by Nintendo? Or is something like this not able to be done?

The facility for this would've had to have been coded into the game on the disc to begin with, and then whatever updates would be stuck residing on the Wii's limited internal 512MB storage.

If the original disc doesn't contain the ability to check for updates in the 512MB flash and use them instead if they exist, then there'd be no update short of a re-pressing of the discs and some sort of swap out (like what happened with Guitar Hero 3).