I'm using a blue tooth Switch controller and its working fine.

I'm playing in Linux (WINE) with a thrid party wireless controller (no issues).
In [Options] > [Controller Prompt Preference], I set it to XBOX.

Sounds like windows is seeing your controllers as XBox controllers, and/or they're setup to use XInput like an XBox controller uses (If you've got like, a logitech 710, you'll see a switch on the back that swaps between X and D -- that's XInput and DirectInput)... a PS4 controller uses DirectInput and requires something like DS4Windows or another wrapper to be able to work (which is why the steam version is unaffected -- it natively wraps inputs)

Does the game allow for keyboard and mouse only input?

Yes.

I think it also works keyboard only, but not mouse only.

Thank you, dtgreene. :)