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 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