>>11980667
>>11981252
They certainly would've helped.
It'd be easy to just say they'd release all those Xbox games as GameCube games instead, but personally I think one or two of them might've ended up getting cancelled or delayed to Revolution/Wii in favour of other projects.
If Nintendo had bought Rare outright, they would've been more directly involved with the studio, and they would've wanted a return on investment. That means new Banjo and DK games would become top priority. I personally think if Nintendo bought Rare then Banjo 3 and a new 3D Donkey Kong for GameCube would've been inevitable.
Then there's Conker's Other Bad Day, which leaked (I think via a trademark) in 2001 before the MS buyout. In our timeline, it was cancelled and ended up becoming Conker Live and Reloaded instead. But if Nintendo buys Rare, there's no need to remake Conker, so the sequel certainly stays in development.
None of this would've been enough to beat the PS2, but IMO a GameCube with Rare on board would have comfortably outsold the Xbox, and as a consequence probably gets more 3rd party support too.