There are arguments that the GameCube can compete with the original Xbox rather well, and I do believe that is true. But I think the GameCube is a case where the developers really have to optimize the machine to get it to do so. The Xbox was quite powerful on a general "on-paper" level.
Xbox was definitely the strongest but also seemed to encourage less optimization (which I say just based on the end results).
There were a LOT of games on Xbox which ran somewhat poorly or felt unbalanced. They might have a cool shader here or there but suffered from other issues. I used to jokingly refer to janky menu systems as "Xbox menus". You know the type - developers would use a Bink movie as a menu background complete with improper frame pacing which would skip and stutter as you transitioned between different menu levels. This was used so often in so many Xbox games.
...but obviously, that was NOT the fault of the hardware at all.
When exploited, Xbox could produce marvelous results and the most optimized games were real superstars. Ninja Gaiden, for instance, is just nuts for the time with huge poly counts, plenty of neat shaders, high res textures galore, a perfect 60fps, huge streamed levels, etc. Sega did a lot of great stuff on there too.
I think some Xbox games pushed things TOO far, though. Riddick is incredibly impressive for its day BUT it made a lot of sacrifices to get there - it was one of the first games I recall with an adaptive resolution but it often dropped way below 640x480 resulting in a very blocky looking game. Performance wasn't that stable either. Or Halo 2 - again, very very impressive tech but the streaming issues were a real problem and the frame-rate just wasn't that great. These were all games that could not exist on PS2 or GameCube, I would argue, but they don't hold up that well today due to those issues.
Those same types of problems plague so many 360 and PS3 games as well. Bad frame-rates, poor presentations, choppy menus, badly compressed videos, etc. Lots of little things that add up to something that feels impressive in its day but ultimately ages poorly.
Games like Metroid Prime, Ninja Gaiden, and Metal Gear Solid 2 are the real super-stars. They exploit the hardware to just the right level producing beautiful visuals while maintaining super smooth performance. They all hold up beautifully in 2016 due to this while, in comparison, many PS3 and 360 games look and run like garbage in comparison despite pushing more advanced tech.
I do feel Nintendo's approach is the right one for console gaming. If you look back at its Wii U games, you can see it all over - so many 60fps or, at least, rock solid 30fps games. They aren't cutting edge but they use the tech to produce highly attractive games that run like a dream. These games will stand the test of time.
A recent example of this is something like DOOM. It's a beautiful game and it runs basically perfectly on all platforms. It feels great to play as well. It is a game, like Metroid Prime and the others, that will hold up in 15 years. id Software absolutely NAILED it in a way that so few others are doing these days.
