Sure you could! I used XNA Game Studio to produce games and apps that would run on my 360. They couldn't take full advantage of the hardware. But if you paid a fee, you could sell your game on XBL much like the App Store or Steam.

https://en.wikipedia.org/wiki/Microsoft_XNA_Game_Studio

It was called XBox Live Indie Games, you could sideload anything you wanted once you paid the initial fee, and then you could put it on the store and accept purchases if it got through a peer review system.