So, after some more digging, I figured out the trick to getting XNA games to run on machines without needing Visual Studio and XNA Studio installed. It seems that the normal DirectX9 package, which I prefer (I don't like web installers), and most people probably used to install DirectX, does not actually install everything that XNA needs from DX. You must use the web installer to be able to run XNA games independent of VS/XNA.
In light of all that, I can now release Asteroid Rage in a form that (mostly) anyone should be able to play! Without further delay, here's what you need...
Beyond that, there's just the game itself--available right here. Enjoy! 