Xbox S|X controller does not work

I've bought a new Xbox X|S controller (model no. 1914). I've successfully used it with my PC, but when I try to use it on my mac (MacOS big sur 11.6.4), it doesn't work properly.

The mac does recognize the controller, and it does connect to it via bluetooth, but when I try to play a game, the game recognizes the controller and sometimes respond to some keys (eg. in the main menu) but doesn't respond to any button in the game itself.

The following notification appears: "This device needs a firmware update to resolve connectivity and performance issues".

I updated the controller firmware using a PC, but it keeps showing me the same message and having the same errors.

When I asked Xbox Support I've been told that it's a MacOS problem - but I understood that Big Sur is supposed to be Xbox compatible (That's the only reason I've updated from Mojave just yesterday!)

I've bought this controller specifically because it's macOS compatible - hope there's an easy solution to this problem.

Many thanks! :)