A 'major' console can play games that the Switch can't. Individual GameCube titles (like Super Mario Sunshine) aren't sufficient; it has to be something along the lines of the NES/SNES/N64 Online collections.
@NcyRocks Is this asking
Will GameCube games be playable on the Nintendo Switch before [Nintendo’s next major console releases]?
Or
Will GameCube games be playable on the Nintendo Switch before [they are playable on Nintendo’s next major console?]
No. Game design has plateu just like any art that matures. Gamecube era games are close enough to modern games to resell them at a high price and be comercially succesful. You can't do that with Nintendo 64 era games for example.
I love the GameCube, and would love them on Switch. I see two potential issues:
Emulating the GameCube well is very hard. Dolphin is an amazing open source project, but it has some great developers and has had many years.
Controls from the GameCube don’t map 1:1 with the Switch - the GameCube has analog triggers (like in Mario Sunshine for example, the farther down you hold the triggers, the farther the water sprays), while the Switch’s analog sticks are digital.
AFAIK the only first party GameCube game to get a straight port to Switch was Super Mario Sunshine for the Super Mario All-Stars 3D collection.