Having lived through every generation: Gamecube is the worst one, the Wii U was also fairly bad, and the N64 is the only other home console I can really say a lot of negative things about.
Problems with the Gamecube:
1. Uninspiring lineup - games had a smaller vision than either the N64 or Wii installments.
2. The kiddy image - while Nintendo had a kiddy image from the beginning, it was always questionable until the obnoxious marketing and presentation of the Gamecube came to light.
3. The controller forced some games to be cancelled (like Street Fighter) and others to have simplified controls with missing features (like SSX). Just try playing Super Mario World, DKC, or Street Fighter 2 on VC with a Gamecube controller, and see how terrible it is. They're basically unplayable
4. Droughts - the Gamecube was a dark age of unimportant games. Generic third party multiplats that were almost always more compelling to play on other consoles (I think Link being in Soul Caliber 2 was the only exception... too bad the controller sucked for that game making some characters virtually unplayable). But even with those, the number of games released by Gamecube was minimal at best with about 100 game releases on average per year (for comparison's sake, the Switch gets about 125 per month) with some months getting fewer than 5 games. And of these 100 games, only a very small percentage of them were of real interest (As far as I'm concerned Animal Crossing being the only one that comes to my mind for the entire generation). Basically, the vast majority of Nintendo fans stuck to handheld this generation because it was too depressing to watch almost every major gaming event of the generation unfold on PS2, and sometimes Xbox... but at least handheld had a lot going on, too. Meanwhile, Nintendo fans had to pretend games like Tales of Symphonia and Pikmin were a gigantic deal... which brings me to my next bit:
5. Overzelous fans overselling the importance/quality of games didn't help, it was mostly grating and embarrassing watching Nintendo fans become very cult-like in this time... over something that was kind of toy-like. These fans didn't want to admit its flaws and instead blamed everyone else except Nintendo for why it was doing poorly. It destroys the credibility of anyone who claims the likes of Mario Sunshine or Mario Kart Cube are the best games of the franchise. Celda was another one, which was slow and doubled down on some of the worst dungeon design from the N64 Zeldas, and while the game was much more animated, it was also unquestionably aimed at kids. Then there's Eternal Darkness, which Gamecube fans hailed as the greatest survival and horror game ever made - despite failing to implement it's "insanity" feature in any meaningful way (it was more of a gimmick) and having some of the worst balancing of any game seen since the early 8-bit era, it was atrocious. Essentially, Gamecube fans were/are so cult-like that they're still defending this turd until this very day and acting like the console wasn't a miserable failure.
While Wii U was a fairly miserable time to be a Nintendo fan, it was coming off the back of a highly successful period, whilst Gamecube was coming off the back of the N64, a console similarly plagued with droughts and missing out on the most compelling experiences of the generation. Nintendo's home console division actually made more money during the Wii U generation than the Gamecube generation - but overall, Nintendo's revenue downturn in their handheld division brought lower net income.
Anyway, overall, this is how I see each console:
NES - Roman Kingdom and Senate run Republic
SNES - Imperial court run Roman Empire
N64 - Late antiquity, general decline with flashes of brilliance
Gamecube - the dark ages, a poor mimicry of the PS2 with zero inspiration. PS2 was the much brighter Caliphate, and the Handheld industry was like the Byzantine Empire.
Wii - The revolution and Renaissance
Wii U - The betrayal of the revolution's values, and a sudden devolution back into a darker ancien regime. Kind of like the reign of Louis XVI or the depressing reign of Oliver Cromwell which at least was short
Switch - The Enlightenment and birth of the modern era, the beginning of a highly productive era for Nintendo, inspired by the Wii and values of its revolution.
I describe myself as a little dose of toxic masculinity.