Thanks for exhaustive reply. OK, the solution is to press Enter to start and use keyboard controls on Xpadder. The mess is that is not possible to use the method for multiplayer becouse the game supports a single player only on keyboard. Well, maybe I have a last chance: "XInput Plus" uses custom editable dlls (xinput and dinput as well). When placing it in games folder, the game's exe read it bypassing the default of Winows folder. For Unity games doesn't work, strangely the beep of detection cofirmation sounds but controls not changes. I tried placing it everyware, also replacing the original on Windows folders, but game do not start in this case. Have you got idea where to put them, please? If not, really thanks anyway!
Ah. Yes, I see how that would be problematic then. I'm not personally sure, unfortunately. My understanding is that the build of Unity we used to compile the game does not play nice with xinput in general, but I also can't really confirm that's the problem. Sorry.