xbox controller works flawlessly with steam under linux. As long as I connect it before i launch the game. A few games pick it up after, but most do not.
Arch based distro here. (steam deck is arch also)
I have no experience with the controller you bought though.
First step would be to open a linux designed controller game, outside of steam and get it working on that. Think minecraft outside of steam.
Make sure the controller registers and works there before steam. if so, the the problem is likely on the steam input side of the problem. IF you cant get the controller working outside of steam in a game either, its likely going to be a driver issue for the device not working.
What I am finding in quick searches about that controller is that people got it working by getting the xbox controllers working.