I was a bit annoyed when my shiny new controller showed up yesterday and didn't "just work" right out of the box. It connected via Bluetooth just fine, but I knew something was a bit fishy when it showed up as a generic controller. Turns out you need to make sure Windows 10 is updated for the controller to be properly recognized. In the Windows update screen, you're going to want to look for "Feature update to Windows 10, version 20H2" and make sure it's installed. I had to get it going manually. Once Windows reboots, add the controller in your Bluetooth settings and voila! Works like a charm.
Well, kind of. For whatever reason in Steam Big Picture Mode, it's pretty wonky for me. Not sure if this is a widespread issue or just me.