Half life 2 on Xbox dips to less than 15fps at points, (Digital foundry video) and doom 3 doesn't have near the polygon counts as some gamecube games (in fact it's just low poly period), and has very low resolution textures. Neither of those games are representative of Xbox's capability, there are more impressive games on the platform, let alone the cube. I remember reading that Carmack statement, he took a quick look at the gamecube and didn't have basic understandings of its tool set. (TEV) Nah, gamecube was just the superior machine. Metroid prime runs at a blistering 60fps and looks better than both, though it may not have the physics of half life or the complex lighting of Doom.
Here's a blog post that I like discussing the gamecube and Xbox.
http://www.purevideogames.net/blog/?p=479
As for the Ps2's cpu,a lot of that power has to go towards graphics, pushing polygons and all sorts of tasks and it isn't directly comparable to a general purpose cpu. Same thing with Cell, if you look at the numbers it seems to be 6x more powerful than 360's chip, but the consoles ran games of very similar complexity. Once again that's because the Cell has to process a ton of graphics tasks, otherwise the system would lag behind the 360 with its superior graphics chip. Not to mention that for both of those Sony consoles, the theoretical capabilities are much higher than the consoles can produce in practice, while Nintendo posted highly conservative numbers for the cube.
Burnout 3 didn't come to the gamecube for one reason, sales. Gamecube got a lot of first in series games, only to miss out on later games in a lot of instances. But Burnout 3 is a very impressive game. As was Black, another game that missed the cube.
All 6th gen consoles had some advantage over another technically, really. The PS2 had monstrous fillrate, the Xbox had programmable shaders and a gpu that in some ways outperformed the cube, Dreamcast had much better image quality than PS2 etc. etc. Overall it's GC > Xbox > Ps2 > DC, but everything after the dreamcast had some extremely impressive games at one point or another.