That's a very clean looking pac stick. Very little rust.
Just at first glace, it looks like you left switch's leafs are not making contact correctly like the other switches. If this picture was taken with the joystick at the rest position, there is a space between the left switch and the actuator (red knob thing). See how the long leaf blades on the other switches rest up against the actuator? When you press in a direction, the long blades push and make contact with the smaller blade to complete the circuit.
You can first try to adjust the blades using a pair of needle-nose pliers. Get the longer blades to rest against the actuator, and adjust the short blade so that when you press in a direction it makes contact together. You can adjust the "sensitivity" of the leafs this way by making the short blade bend closer to the long ones.
If this doesn't fix the problem, you might have to trace the wire using a multimeter's continuity setting to check to see if there isn't a break in the wiring.
It could be deeper than that too, but lets hope what you're describing is just a matter of bending the leafs a bit.