A little additional information: I tried plugging in a several year old LG USB-connected controller. Again, Windows was able to recognize and calibrate the controller (this time, the right stick was the Z axis). Jedi Fallen Order wasn't able to recognize the controller at all - not the buttons, not the D-pad, and neither stick.
Another update. I managed to connect with Origin support, who stated that this controller is not supported with the game. Only Dualshock and Xbox Controllers were indicated as natively supported (not clear if this is a complete list).
It boggles my mind how the OS could be able to interact with the controller perfectly fine, but Origin's software can't.
