I know this is a wii-u discussion thread, but I have to ask. Wasn't texturing one of the areas the xbox surpassed the gc in? Most GC games, even the good ones seemed to have relatively poor textures, whereas many xb games had pretty sweet, high res textures along with stuff like bump mapping and normal mapping.
That was due to RAM and storage constraints, GC had 24 MB of useful RAM (the other 16 MB were slow as hell, they were mostly used for sound and light tasks) and 1.4 GB storage per game disc, Xbox had 64 MB, albeit not all would be accessible and up to 8.5 GB on DVD for storage. In the end, it would still have roughly double the graphics memory going for it, if need be. Although that memory was not as good and part of the bottleneck against how the GC was engineered.
Stuff GC did, like Rogue Squadron 2 was never surpassed by Xbox and that entails stuff like bump mapping on every surface, with 15 million polygons @ 60 frames per second. Best Xbox did was 15 million polygons @ 30 frames per second on Rallysport Challenge 2.
It's really like this; Xbox Geforce 3 was technically better on paper, it certainly pushed more RAW polygons (although Microsoft's figures were inflated) and Gamecube's raw polygons (untextured and not lit) weren't anything to write home about; thing is, the thing had very little hit texturing them and applying effects; and on top of it it also did double the work per pass, it's the story about predictable performance and how much Nintendo loves that, and they hit the nail on the head regarding what would be the best thing to have to make a difference that gen.
In the end, it would trump the Xbox easily on attainable polygon count even without the polygon trick going on over on Xbox; and it had ease texturing and going insane on effects based on EMBM (Environment-mapped bump mapping) problem being... at the same time it didn't have much memory for them, so you could end up using low-res maps, like on Pikmin games, lots of shiny surfaces, but crude.
Then there's documentation, Xbox being standard and GC being pretty non-standard, each had it's quirks, I've read some stuff a few years and I forget specifics, but I know GC had less hit doing a specific kind of bump mapping and Xbox had less hit doing the other; that means if you were to port one game from the either system and not tweak those specifics both platforms could suffer so you had to take it into account. I also reckon GC DOT3 implementation was somewhat incomplete. It would work on a basic form (and be more costly than on Xbox), but stuff like normal mapping wasn't in. That doesn't mean it couldn't do normal mapping (in fact Wii did it in a few games, and that's a GC with 50% more clock), but it certainly wasn't on spec so developers didn't rely on it like they did on Xbox (they also relied on it on Xbox because they had to be pulling less polygons than what they wanted). Anyway, going back to the standard versus non-standard, Xbox was widely accepted to have a work pipeline that you ought to master, pixel shader knowledge/language being relevant to this day, with GC you had to really master it through limited means by comparison, you couldn't get a PS1.1 shader and simply drop it in like so many games did on Xbox, some developers didn't take it as a challenge but simply as a useless exercise and that was the wrong mindset.
As lazy as that sounds, though, on GC if you wanted to preview a TEV pipeline "shader" you had to tape the game out, put it on a console, I reckon only in 2008/2009 did Nintendo implement a TEV pipeline previewer on the Wii SDK. It really did take more time and was less wysiwyg to work with; that somewhat excuses a lot of lazy devs in the realms of, perhaps they really didn't have the time and resources.
In the end, it's basically this: you couldn't pull Halo 2 on GC without significantly downgrading textures but it could pull 60 fps no problemo (on the Xbox it was 30 frames per second endeavor); then again Metroid Prime 2 on Xbox could never hope to go past 30 frames per second on Xbox. Choice software ended up chasing the platform strengths and trying to hide the downlows for both consoles; but the GC wasn't really at disadvantage, no; it was crippled on RAM, disc storage and framebuffer RAM though. (I don't understand to this day how they didn't spruce up the embedded 1T-SRAM on the GPU for the Wii, we would have seen AA a lot more frequently that way; and less dithering too)