The documentation is for Mathematica
The documentation provided by the Programming Cloud "Help" menu is titled "WOLFRAM LANGUAGE & SYSTEM Documentation Center". I see nothing to suggest it is for Mathematica. If you have a source, please cite it.
Most of the things not included in the Programming Cloud
ControllerInformation is included in the Programming Cloud. The docs http://reference.wolfram.com/language/ref/ControllerInformation.html?q=ControllerInformation do not mark it "not supported in cloud", and Programming Cloud does execute it and return a result.
are things that aren't appropiate for a browser application. In this case, the
joystick probably isn't very useful for a browser application where most of the
computation is done server-side.
Gamepad is fully appropriate for a browser application. The docs say the gamepad is supposed to operate the in-browser sliders. The gamepad has no more to do with the server-side computation than does for example the mouse, which works fine.