Come to think of it, wouldn't burning Little Chicago have released all the stored up energy inside? Wouldn't that be equivalent to detonating several kilotons of TNT? I mean, Vittorio's little gauntlet was set on fire, and that exploded violently. The description of Little Chicago's energy reservoir leads me to believe it's exponentially larger and more potent. Hm.
Mostly, I think LC had to go for plot reasons; it was simultaneously important for Harry to do a few things, but arguably too powerful to keep around. With enough practice, Harry wouldn't need to go running around every chapter to hunt down clues. It's similar to why he can't use computers or cell phones (plot-wise, not in-universe).