The original Xbox never got a South Park game, but if it had it would have looked something like this.
The raw work in progress was allegedly discovered on a debug original Xbox procured in August by Nintendo Age forum moderator qixmaster, who posted a few screenshots and a single cutscene from the game.
Sometime between then and now the folks at HappyConsoleGamer got hold of the console, and after a bit of exploring put together the following video. It features the South Park kids exploring a virtual city filled with worded placeholder textures, open road driving, and even a combat test section in which Eric Cartman beats up wild dogs.
So it's basically South Park meets Simpsons Hit 'N Run or Grand Theft Auto. I can't find any indication that the game was ever a proper project, much less why it was canned. We were probably better off waiting for The Stick of Truth anyway.
The game was supposedly under development by Oregon's Buzz Monkey Games, a studio funded by former members of Dynamix. The studio worked on licensed properties and co-developing and porting games in the Tomb Raider series. After a brief dip into mobile gaming the studio was purchased by Zynga in 2012, becoming Zynga Eugene.
Hats off to the folks at HappyConsoleGamer for digging into the original Xbox and pulling out this footage.