I think the hardware is the same basic architecture, so it downclocks to run Gamecube games natively. N64, SNES, and the rest are handled through software emulation.
And I don't think they can patch in Wii controls into a straight Gamecube game for this reason (but I wonder if you can use the Classic Controller?), however, Miyamoto recently spoke about potentially revisiting some Gamecube titles, reworking and revising them around Wii-specific control methods when it would benefit them, and releasing them at budget prices. Not pretending they were Wii sequels, just like they were Gamecube+ games.
I doubt they'll actually do this, though, or at max for just two or so titles. Reactions will probably be bad to it. But hey, they could go back and fix Sunshine up a bit...