How can I start a GameCube game without using a Wii Remote?

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).