I've got no experience with GameCube internals, but I'd check voltages from the power supply. Also, caps can go out-of-spec even without looking bad. A Pentium 4 PC of mine had an issue like that. As soon as they got a little warm, boop! All gone.
Amiga 4000: Apollo Turbo 040, 128MB Fast, 40gb HDD, CD-ROM (needs internet)
C64: 1541U-II, MixSID, WiFi, LumaFix64
ZX Spec, A1200, CD32, MacPlus/SE, A500, BBC Turbo Master (NASA), 2600, Acorn 4000, VIC-20, IBM 5155, GameBoy