What factors do you believed caused the Gamecube to fail though? What boxes didn't it tick?
Since the main pillars (what sells the hardware) of every Nintendo system are Nintendo games, one only has to evaluate Nintendo's own output for the GC because that alone is already so damning for the system that other things don't even need to be looked at.
1. The launch lineup fell flat. While Wave Race: Blue Storm was a great game, the appeal of racing on water is limited. It's so limited that trash like Cruis'n USA outsold the excellent Wave Race 64 on the Nintendo 64. The other Nintendo title at launch was Luigi's Mansion which didn't look exciting because it wasn't exciting. It's a good game on its own, but merely being good doesn't cut it when hardware needs to be sold.
2. The sequel to Super Mario 64 went its own way by giving Mario a waterpack and consequently a water-focused level design. Water levels do not belong to the favorites of gamers; water levels exist to add diversity, they aren't something that a game should be built around.
3. The Legend of Zelda didn't fare any better with its choice of cel-shaded graphics and great sea scenario. Another major sequel that failed to satisfy the expectations of fans.
4. Mario Kart introduced a dual-driver mechanic which hasn't been used since. There were also character-specific super-items that weren't balanced. That's the third sequel that Nintendo messed up. Super Mario 64 sold 11m copies, Ocarina of Time 7m and Mario Kart 64 9m. You aren't going to convince the people who bought those N64 games to get a GC when the sequels stray off the path so much.
5. Nintendo's bread and butter of the 8- and 16-bit eras, the 2D platformer (most notably Super Mario Bros. and Donkey Kong Country), was missing in action once again. The Nintendo 64 could limit the damage of this oversight by wowing players with successful transitions to 3D, but ultimately 3D is not a proper replacement, so 2D titles still need to be made. Nintendo made Donkey Kong: Jungle Beat for the GC, a 2D platformer that was controlled with the bongo peripheral.
Super Smash Bros. was the only major IP that Nintendo improved upon in the correct direction. Melee ended up as the bestselling game on the GC.
Any new IPs Nintendo made fell into the same category as point 2-4. It didn't look like Nintendo was really interested in making games that people wanted to buy, instead the developers got to do as they pleased, getting a free pass to (completely) ignore expectations.
The bottom line for the software lineup is that it was received with a big fat meh by the market. It didn't take long for the GC's price to drop to $99, but people still didn't want to buy it. The GC hardware design was okay for the most part, although the controller made it clear that 2D platformers wouldn't return. The games the controller was made for underdelivered, the notable exceptions being SSBM and the Metroid Prime games. Metroid's appeal is limited though; not only does the isolated sci-fi theme remove virtually all females from the potential audience, but it's also a difficult game with no handholding.
The GC was missing on the big hit games. An analysis of a Nintendo system's game library isn't as simple as saying "this console had Mario, that one had it too, so this can't be a problem." It's a bit more complex than that. The first thing to look at is if big IPs are represented, the second thing is to evaluate if said IPs actually lived up to expectations. Hardware-wise, processing power doesn't matter, but controller design does. Because the controller has a big influence on how games are designed and which games are being made to begin with.