I, too, have a broken GC that can read disc for a couple of seconds before the whole drive stops spinning and the "no disc" label pops up. The odd thing is, I can use it for Animal Crossing just fine. But that isn't the only thing I use the broken GameCube for. I also use it as my new "GameBoy Cube" system(don't know what that means? figure it out for yourself). Sometimes, I would also get the "no disc" label, but on rare occasion.
Since I don't have a working Cube right this moment, I'm patiently saving money to get the new GameCube/Zelda bundle when it comes out this holiday season. When that time comes, I will have the best of 2 worlds, a GameBoy console and a GameCube console. Damn those tactics games and those SNES remakes! I'm going broke!!!
My advice to you... keep your old GC and just get the new GC/Zelda bundle when it comes out.