City Folk is by and large considered the worst in the series, so I think any Switch game you're considering getting will probably be a better investment.
As far as GameCube games go, the original Animal Crossing is obviously a good one. Luigi's Mansion is a bit of a divisive game, but I think it's a lot of fun, and it seems to have a much larger cult following in recent years since Luigi's Mansion 2 and 3 have expanded upon the formula. If you're into RPG's, Phantasy Star Online, Digimon World 4, and Tales of Symphonia are all excellent picks. Metroid Prime is one that I'm constantly being recommended to the point where I picked up the trilogy on Wii, but haven't gotten around to playing it. Sonic Adventure DX is a flawed but serviceable port, Sonic Adventure 2 Battle is also good. Pac-Man World 2 is excellent and underrated. Super Mario Sunshine is another one of those games where I understand why some people have a kneejerk reaction to it--because it has quite a few segments that are absolutely terrible--but I feel the overall game is still very fun.
Super Smash Bros. Melee is another one people will probably recommend the heck out of it, and while it is a game I have a huge amount of nostalgia for (and I'd argue it has the best overall aesthetics, atmosphere, and music of any of the games), it might be difficult to go back to if you've played any of the more recent Smash Bros. games. Soul Calibur II is a good fighting game and the GameCube version has Link, which is a bonus. Wind Waker is good, but I think the Wii U version is probably better. There are some people that vastly prefer the GameCube version of Twilight Princess, so that's worth looking into as well. The GameCube Pokémon RPG games are also fun for what they are, but not a substitute f or the handheld versions, which is ironic since usually it's the other way around.
There's probably more I could list, but those are just the ones off the top of my head. Good luck!