The Nintendo Wii is one of the 3 "main" consoles people play today, and the online support for it is horrible. If you have ever tried to play Super Smash Bros Brawl online, and the other 3 people aren't your next door neighbors, you will feel my pain.
I think Nintendo has focused more on great single player games, think about it. Ask a random person to name their favorite Nintendo games, modern or retro. You will probably get games like Ocarina of Time, Mario Galaxy, Final Fantasy 3, Donkey Kong. If you compare that with favorite Xbox or Playstation games? Halo, Gears of War, MAYBE Uncharted, Call of Duty, Battlefield.
I'll admit the Playstation has been around longer so it has had more single player games, but Nintendo is the grandfather of them all, and I think people go to Nintendo for offline gaming mainly, and that just isn't going to change anytime soon.
If you ask me, online gaming is best done on the PC. The ease of programs like Ventrilo and Teamspeak, with all the games that are available on Steam for instant download, and the amazing hardware capabilities that will ALWAYS zoom ahead of consoles?
However, I still prefer offline gaming. Offline games are relaxing, story-driven, and while they may not have the same replay ability as multiplayer games, I never get turned off of them by other gamers. I used to play WoW, but other gamers really turned me off of it. I still go back for the occasional Mega Man 2 playthrough.
Sorry, I know this wasn't really about the Gamecube but it sort of ties into it.