Is it possible to just use the Wii as a GameCube without the need for a Wiimote (using only the Wii console and the GameCube controller)?
If not out-of-the-box, what settings should I set, and will I need a Wiimote to set them?
No. You cannot select the game in the Wii home screen with anything but a Wii controller. Alas, of the 12 entries in the Wii's settings, none has anything to do with the controller, save for setting whether the Wii sensor bar is above or below your display. This question is rather common, but given how newer Wii no longer have support for that hardware, it does not seem like Nintendo intends to add this option in.