It's a bit more complicated than that (on Windows) because Steam doesn't make a virtual gamepad to the OS. The way Steam handles the input is by hooking into the games individually. So to use Steam for other games, you need to add them to Steam as non-steam games.

Even open source controller remapping tools (not just Steam Controller) and similar used ViGEmBus which is no longer maintained. You can have it do mouse/keyboard though, those don't require custom drivers.