XBox One S controller (white) connects but won't transmit button press or stick movement ...

Try doing a hard RESET on the console EXACTLY like this. Hold the power button on the console for 10 seconds or until the power shuts down completely. Unplug the power cord from the back of the console for 5 minutes. Plug the power cord back in and restart
the console. Once it is back up and operational test the results. Let us know. Good Luck!

the Left stick is the one by the D-Pad, and is usually used to control character movement within a game.

the Right stick is the one by the buttons and is usually used to control the camera/view/aim within games

hope this helps

Please test the following:

Open the Start menu and type "Control". This will bring up the Control Panel (or localized name for the Control Panel) in the search results.

In the Control Panel, type "Game" in the upper right search field and open the control panel for Game controllers.

Please check that the test tab reacts to the stick movements and button presses of your controller. Does every button and stick work as intended?

In the game, are you unable to use the controller in the game at all, or is it just that you can't change the button bindings in the game settings?

I found one. Chime in if this does not match with what you know.

Xbox One controller layout

Combat

Left Stick - Movement

Right Stick - Aim

Left Stick press - Tactical Power

Right bumper button - Class Mechanic

Right Trigger - At Will 1

Left Trigger - At Will 2

X button - Encounter 1

Y button - Encounter 2

B button - Encounter 3

Left bumper button + X button - Daily 1

Left bumper button + Y button - Daily 2

Left bumper button + B button - Artifact

Non-Combat

Left Stick - Movement

Right Stick - Camera

Right Stick press - AutoRun

A button - Interact

Left menu button - Focus on status

Right menu button - Start menu

D-pad up - Mount

D-pad left - Consumable item 1

D-pad down - Consumable item 2

D-pad right - Consumable item 3

Left bumper button + D-pad up - Map

Left bumper button + D-pad left - Inventory

Left bumper button + D-pad down - Chat

Left bumper button + D-pad right - Invocation

Left bumper button + A button - Jump

It's probably your controller.

The black controllers are not as well made as the old white ones.

Every single black controller I own has one of the sticks that isn't centred properly so some games veer to the left or right?

It doesnt effect all games but effects a lot. Depends just how sensitive the game is to movement of the sticks.

Either I'm extremely unlucky or the controllers aren't as good as they used to be.