It's a bit more tricky, a Generic HID just gives you a DirectInput device, while reasonably modern games use Xinput. Microsoft never provided a way to map DirectInput devices to Xinput. For Xinput to work a Microsoft specific USB protocol is needed, not a Generic HID device. Many third party controllers have a switch or button combination to switch between XInput and DirectInput modes for this reason.

Microsoft has a new API with GameInput that addresses this situation and allows mapping Generic HID devices onto game controller via config file, but it doesn't work retroactively, it only works for games that use the new GameInput API.

Valve could of course provide a way to switch and emulate other protocols too, just like other third party vendors do, but there is no USB standard that makes things "just work" in Windows when it comes to gamepads, you always need extra drivers, USB modes or other hacks.

On consoles the situation is even worse, modern consoles deliberately lock out any unlicensed third party controller. Playstation3 was the first and last console that supported standard USB controller, while PS5 doesn't even support PS4 controller.

2 thumbsticks is already 4 axes. Add 2 triggers and it's 6 axes. Add gyroscope and it's 9 axes. That's more than 8. And I haven't mentioned the touchpads.