I can sometimes receive input from my Nintendo Switch Pro Controller, but then it stops working quickly and most of the time it won't even work at all. I use Unity 2021.2 on a Windows 11 machine (but I had the same issue before updating from Windows 10 to 11).
What I mean be "receive input" is that I can go to the Input debugger window and see events from the controller. Also, I was sometimes able to assign a button to an action in an Input Action Asset using the "Listen" functionality.
Outside of Unity the controller works perfectly (e.g. in the Windows menu). I have reinstalled my bluetooth drivers today, which did solve connectivity problems that I was having, too, but the problem described here remained.
Here are screenshots of the windows where I would expect to see my inputs:

When I click on "Record Frames" I start seeing events in the bottom pane, but they appear to be from a different device (Device ID is 0, not 4).
And this is the "Listen" button I was talking about:

I scratched out the keyboard keys, because those were triggered by me making a screenshot. When pressing a button on the gamepad nothing shows up.
I haven't used the new input system before, neither have I ever successfully used a controller with Unity - so, don't assume I have tried the "obvious", since it might not be obvious to me :-) Thanks for taking the time to read!