I would put the would put the wii firmly in the gamecube family line. it's a uprated gamecube with a weird controller.

    nes:snes = 6502
    n64 = mips
    gamecube:wii:wiiu = powerpc
    switch:switch2 = arm

Yes, the Wii is essentially an overclocked GameCube with a bit more RAM and as you mentioned, a weird controller.

As a regular human who plays games and doesn't know about chip architectures, one woud probably lump the wii and the switch closer together than the game cube based on the modes in which one can interact with the systems.

Wii is a game cube with a funny controller. Or, wii is a tv-only olde switch.

I appreciate that it has its own name due to being a transitional experience.