Even the Wii itself will run Gamecube games in Wii mode, see "Nintendont".

I was going to say that the GameCube and Wii are very similar to the point where the Wii is backwards compatible but I guess that argument would apply to the Wii U too. Either way, Dolphin doesn’t emulate the Wii U

Yeah, Wii-U has extra hardware for backwards compatibility with the Wii. Wii->Gamecube has no extra hardware, just the ability to disable parts that the Gamecube does not have.