The problem with Game Cube was:
1) Nintendo had no hype for this machine. Sony and Microsoft are PR masters. They got everyone to get excited about their systems. Nintendo was quiet. I wouldn't doubt most people had no idea the GC existed until they saw it on the shelves.
2) Their trademark franchise games all fell short. The big thing that worked for Nintendo for the N64 was that the system had few releases but the Nintendo releases were excellent and worth waiting for. Everyone waited for the new Mario, Zelda, Mario Kart, and got burned. Wind Waker, Sunshine, and Double Dash are the worst in their series. This caused a lot of people to become disillusioned with Nintendo. 2003 seems to be the year of doom for Game Cube, probably the time where most consumers realized GC was irrelevant in the scheme of gaming and that Xbox and PS2 offered much more. Even in the areas Nintendo specialized in.
3) Even worse post launch schedule than the N64. PS2 was really picking up late 2001 on and Nintendo had nothing to stop them, except Resident Evil, a series that most people have moved on from. Nintendo pretty much allowed Sony to dominate and Microsoft to chug ahead.
4) Nintendo's poor strategy of hyping their product. Their "NDA" strategy worked for them in the past but blew up in their faces this time around. There's just too much going on in the industry for Nintendo to keep quiet. Instead of being anticipated and mysterious, they were forgotten. By 2003, especially once Wind Waker was released, nobody really considered the Game Cube a threat anymore. Xbox and PS2 moved right along with Mario and Zelda released. It wasn't going to get better.