There's no reason why you can't be the first to do so! I think it's just a coincidence that none of the past or current runners used the Xbox version.

I've been told that running the GameCube version of the game on a Wii results in the fastest loading times, and I have seen proof that GCN is faster than PS2, but I have never seen what it looks like on Xbox. Record a run and we'll find out!