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.
If you're OK with installing the homebrew channel, there's an unofficial GC game booter for it. If you make your wii start in the homebrew channel by using bootmii and install the booter, you won't need a wii remote to start a GC game as the channel can be controlled with a GC controller.
However the GC booter doesn't work with all games, and of course you'd void your warranty (although I don't think even Nintendo cares about the Wii nowadays).
You can't use a GameCube controller for your Wii, but the Classic Controller or Classic Controller Pro can perform almost all the same functions as the Wii remote. Plug one of these in your Wii remote, and you can use the right joystick to launch the game or other actions which require pointing with the remote.
With this, however, there are three problems:
But, according to my knowledge, it's the only Nintendo factory solution for a more convenient alternative of the pointing Wii remote with Sensor Bar.