In the latest update for the Nintendo switch there was no mention of GameCube controllers compatibility but obviously, support has been added. Maybe Nintendo just needs to get better at writing the changelogs.
For this to work you will need a USB adapter that Nintendo released for use with Super Smash Bros. for Wii U. Once connected they will show up as USB controllers but only GameCube controllers are supported for now. This has been attempted with other USB controllers but with no luck.
— Master Mewking (@MasterMewking) October 23, 2017
If the game is playable without using ZL-equivalent shoulder button and the home button the GameCube controllers will work flawlessly. Here is a video that explains this in detail.