If you hack the wii and use nintendont to load your gamecube games, you can play the games with a wiimote and nunchuck or the wii classic controller plugged into the wiimote as well as other usb hid compatible controllers.
Without using home brew you can only play with a gamecube controller.
Set your controller to enable pairing (For PS4 controllers, hold down the PS button and the Share button until the light on the rear of the controller).
Click the Apple logo in the top left corner of your screen.
Select System Preferences.
Hollow Knight seems to have an issue with Xbox controllers in general. I've done some research and found the following articles:
Right click steam and open up "big picture". Once there, click the "gear" in the top right. Open up controller options, and enable xbox 360 ...
There could be any number of things that could be contributing to this issue.
As you mentioned, you have tried the default options, of resyncing, resetting, etc. Some other options might be:
1. (as provided by Wondercricket) Clean your Controller:
sounds like you might have a sticky or malfunctioning button or trigger, which is overriding the input ...
When you are playing a local multiplayer game, you use a different account for each DualShock. I think the purpose of turning off the controller is to keep playing singleplayer. Because otherwise, it wouldn't be possible to sign out from the second player's account and turn off the controller at the same time.