OK so you have very thick cables there and you heated up the soldering points a lot they are dark brown the pcb is quite toasted. You can try thinner and for testing a very short cable, rip appart a Network cable for example. I think this will not solve the problem. I think you damaged the contact plates and if you just move the whole thing a bit its disconnected. The right left movement is done by the upper potentiometer and the forward back with the left potentiometer if you look from above.
Resolder the stick, get a new controller and do not desolder the joystick from the controller pcb but just remove the drums and cut the line in the potentiometer box as I showed.
If it starts moving it means the 5k resistance is gone in one direction so the middle pin has no 5k resistance to the side pin.
Try other wires if that is not working you damaged the controller pcb. If you resolder the joystick the plate seems to have contact again .
For example Arctics controller pulled the trigger all the time. Same issue, I seem to have damaged the contact point when resoldering, I was lifting the contact plate as I did not desolder it enough. I was able to solder a bridge to the pcb but it was not stable enough so we took another controller. The new controllers with the shiny X are worse in quality sadly.
SO to sum up I think the problem is a PCB board contact.
