Yes, I run my copied GC isos on my softmodded Wii. Later edition Wiis don't have the joystick ports, and won't load the mini discs, but my early edition works fine.
The Wii is basically an overpowered Gamecube with slightly better video out options. I made a mistake in my post above; the Triforce board was the arcade version of the Gamecube; I'm not sure if any arcade games were made using the faster clocked Wii chipset; the Triforce was a three way release from Nintendo, Sega and Namco, and after it Sega went x64, and Namco went Emotion Engine (PS3 type, I think) based until they too went x64.