Just a quick correction, the Wii was still PowerPC based like the GameCube. It just took both the GPU and CPU and upped their power a bit, but was not different from the GameCube in any real significant way.
The Wii did have a few distinctions from the GameCube. However they abstracted them in an interesting way
The Wii itself was still a G3 PPC chip, just add a few hundred Mhz
It also had more RAM (which allowed the "Twiizers" attack over a decade ago)
All the Wii-centric functionality (USB, Bluetooth, Wi-Fi, SD card, etc) was abstracted away inside an ARM co-processor inside the GPU. Named "Starlet" by the homebrew community
Wii software was then written that would simply poke at the API's it exposed to add that functionality
Ah, yeah totally forgot about starlet. It's been a while since I messed around in the Wii homebrew scene, but that was basically got me interested in cybersecurity in the first place when I was in middle school. I should go back and look at it now that I actually know what I'm doing (more or less).
The Wii itself was still a G3 PPC chip, just add a few hundred Mhz
It also had more RAM (which allowed the "Twiizers" attack over a decade ago)
All the Wii-centric functionality (USB, Bluetooth, Wi-Fi, SD card, etc) was abstracted away inside an ARM co-processor inside the GPU. Named "Starlet" by the homebrew community
Wii software was then written that would simply poke at the API's it exposed to add that functionality