The CX-40 is very simple inside, to be sure, and incredibly robust. It consists of a single printed circuit board (PCB) with five contact switches/traces and a ground plane. These are for up, down, left, and right joystick directions, and the fire button.
Normally, these switches are connected directly to the 2600 console by plugging in the cable which is wired to the PCB. Each switch, when pressed, closes a circuit from it's pin to ground.
Let's open it up and take a look inside.
Use a Philips screwdriver to unscrew the four screws from the bottom of the joystick.
Then you can pull open the joystick by carefully prying up the top.
There's a really good chance that 30 years worth of dust and grime has accumulated inside of the joystick. You may want to take this opportunity to wipe that off.
This is the PCB. Each one of those little metal domes is a button, which closes a contact to ground for when pressed.
We won't be using these wires so you can go ahead and take some pliers to pull the crimp connectors off of the board.
Then you can remove the whole cable assembly entirely.