Good point! I say "gamepads" as any controller with analog triggers basically. I'm not sure what better term to use there. And you're right about the different inputs. Most controllers should still work in the game, but I just set the standard xbox buttons in the description to reduce clutter.
I tested this with a bunch of controllers. And, indeed, there is a grim sense in which the game "works with every controller" - it will always force a layout. I don't mind that, per se, except that I can't disable it. If I could, the game would be become controller-friendly on an even higher level.