I think that your Wii console will need to be repaired, one thing I would check is whether or not there might be a problem with either your GameCube controller(s) and/or GameCube memory card(s), and seeing if by trying another controller or removing the memory card(s) might correct this problem.
Not all Nintendo Wii console that are black lack the Nintendo GameCube support. Nintendo released the Wii Sports + Wii Sports Resort and Mario Kart Wii bundles, which feature the original model Nintendo Wii console that can play Nintendo GameCube games in your choice of black or white. There was also the limited edition Mario Red Nintendo Wii console, which features an original model Nintendo Wii in a special red color.
The only Nintendo Wii console that for sure will never play Nintendo GameCube games is the limited edition blue Nintendo Wii console.