Hi, I was around as a gamer back in the time when the GC was released.
Back then, Nintendo had lost a lot of momentum when they released the N64. Sure, it was a very powerful console if compared to the PSx, but the proprietary cartridge was expensive and was a barrier to entry for many smaller companies, while the CD-based games were cheaper and also had much bigger storage capacity and were better for storing music. They also burned a few bridges with some of the bigger companies, like Capcom and Square, at the time.
Not to mention that during the NES and SNES era, many companies were strong-armed to make their games console-exclusive, which also diminished their profits by releasing on only one console.
So, when the GC rolled around, they were rebuilding their relationships with the third parties, but they still did the whole proprietary storage that gave Nintendo a bigger cut, if compared to both PS2 and the original XBox that used regular media. The storage was also smaller than the others, which also contributed a little.
Nintendo also made a point of firmly staying in the kid-friendly zone (even with outliers, like Eternal Darkness, Resident Evil 4 and Metal Gear Solid Twin Snakes), while the other two were focusing on teen audiences, who had slightly more freedom in getting their own games, and young adults.
And it was also when the third parties that Nintendo had burned the bridges started to try to get back to a good footing, like how Square released FF Crystal Chronicles. They still had the advantage that the XBox executives f'd up big time when trying to get the support of the Japanese companies, so they were more into making multiplatform games on the GC than with the Americans who told them to their face that the Japanese way of doing business was no good and they were going to show them how it is done (yes, it was a real thing).
And there is also the hardware. The controller wasn't exactly very ergonomic, so it could be uncomfortable for some. All the GC accessories had to be Nintendo-approved or first parties, so they were expensive, besides other ridiculous ones that were far more expensive than they should be.
This is why the third-party support for the GC was so underwhelming in its majority. Sure, there are very good games, but not as many games as the others.
Just look at the Wii: it was when Nintendo finally used the same media as everyone else. Sure, the console got tons of shovelware, but it also allowed smaller companies to release many experimental games like how it was with the PS2 and XBox. I see the Wii as an extension of that era rather than a direct competition to the PS3 era.
Sorry I rambled, I tried to be as impartial as possible here.