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.