GC had DVD, Nintendo still didn't care about 3rd parties, so not much changes again.
Nintendo cared, courted, and publicized how many third parties were onboard with the GameCube. They claimed they learned their lesson. Even got the famous "Capcom 5" exclusive games that eventually became the "Capcom 4" because of poor GameCube sales and then 3/4 of those wound up going to the PS2 because it was such a Juggernaut.
With the Xbox being more powerful and the PS2 being more popular and both having better storage (just like with the N64, the GameCube version of games would be missing cutscenes, levels, music, etc), there was just no reason to support the GameCube over the competition.