This is such an easy answer; Xbox's library in America catered to the developing tastes of the time, generally speaking. Not just in having more mature games, but in specific genres like FPS, WRPGs, racing games etc.
It's a moot point to try deciding if one's library was
better than the other's because that type of judgement is almost always subjective, and in terms of technical measures there were graphically intense games on each system some that in several ways bested the PS2 versions on a technical level (Splinter Cell for Xbox, RE4 for Gamecube).
While the Gamecube was more dissimilar to PS2 than Xbox, at least on the surface, when you actually peered at them a bit deeper it's likely the Gamecube had more in common with the PS2 than Xbox in terms of Western support and what Japanese games got ported over and that could have negatively impacted it. Xbox got a lot of unique Western support in games like Morrowind, Jade Empire, Indiana Jones, Riddick, DOOM 3, and KOTOR that simply weren't on PS2 and didn't really have equivalents there. So I think a lot of PC people in America who were beginning to migrate to console at the very least went with an Xbox in addition to a PS2 due to those reasons.
OTOH, a lot of the big games Gamecube had that gen, had counters of comparable quality on PS2. Sunshine? Well you've got Jak & Daxter, Sly Cooper, and other 3D platformers on PS2. RE Remake, Zero, and 4? Well PS2's got Silent Hill, Fatal Frame, and goodness knows what other survival-horror games. Metroid Prime? PS2's got Killzone, BLACK, TimeSplitters (I know some of these were multiplat but they were strongly associated with PS2 so them being multiplat did not do Gamecube or Xbox many favors), etc.
Nintendo's own software brand did very well that gen (even if several games from them underperformed compared to internal expectations), but they lacked the Western 3P support long-term, particularly in the form of exclusives, to help drive Gamecube in NA and Europe the way the N64 benefited from during its era. There's really not much else to it. Sure the $99 price cut might've been seen as a sign of weakness but they only took that option because Western sales were already lagging behind PS2 and even Xbox, and the gap was already growing. The price cut was Nintendo's way of trying to stop the bleeding, it didn't work.
Xbox sold 20% more than GameCube, that's not close.
Also kinda this. Maybe very early on they were close but I'd define "close" as within some margin of error percentile, I'm talking 3% - 4%.
Maybe 10% max.
But Halo was a massive launch title for OG Xbox at the time and the Gamecube really had nothing to counter it, so whatever gap was there between both systems in NA at the start grew in Xbox's favor relatively quickly, I'd imagine.