The Wii was it's own console because of it's controller. If the PS4 is different from PS3 because of more powerful hardware, the Wii is different from GC due to the controller. And DS is certainly not GBA. And the Wii U internally was very different from Wii.
Interestingly there was a recent leak that showed the Wiimote prototype was indeed originally an accessorie for the GameCube.
They just decided to repackage it and make it a new hardware platform, but I recall during the GameCube era Iwata was saying they had some new accessorie coming that would expand the appeal of the GameCube, this was pretty much it.