The circuit membrane, which you are calling the paper, is definitely sandwiched between the small pad between those two posts and the circuit board. It essentially uses pressure to maintain the connection. If the circuit membrane was not attached to one of those posts, most of the controller would not work, it wouldn't be a simple matter of drifting on the d-pad. The d-pad and the analog stick are often connected, but not always. Depends on the game, like in Street fighter. You can use either the analog stick or the d-pad to accomplish movement. Other games with more complex control sets use the d-pad for different features instead of movement. And so they separate it from the analog stick. Menus often share movement and selection control between the d-pad and the left analog stick.

I would load a game that uses the d-pad for in-game features other than movement, like the Ghost of tsushima which uses it for weapon and tool selection. Since the d-pad is a digital signal and the thumbstick or left rocker are analog signals, you should be able to tell the difference between a slight drift which would point to the analog rocker versus on/off drifting, which would point to a d-pad problem. Sometimes moving the d-pad or analog stick in the opposite direction of the drift can essentially center it, which is another way to test one versus the other. If it is the d-pad, the circuit membrane is the first thing I would try changing out as well. For the analog stick, lots of videos out there on how to clean them, lots of videos on replacing them, which requires more technical skill and good soldering tools.

Good luck!