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Testing the Pot
Your potentiometer appears to have corrosion. If this is malfunctioning, it could mean the laser has been firing without a limit. Can you use a multimeter to see if you get a readout when you test the 5V and G pins? It should read within a few ohms of the rated resistance of your potentiometer.
What results do you get if you test between the wiper pin and one of the other pins, rotating the pot knob slowly from one side to the other? One end should have only a few ohms and the other maximum resistance. It should slowly and continually decrease/increase as you turn the knob with no sudden jumps.
The laser power supply unit (LPSU) in your images has corrosion on the screws and a considerable amount of debris around the fan opening. If any of that has gotten inside of the LPSU, it could potentially cause performance issues or worse.
Shorting the Connection
We are going to replicate how the board “talks to” the LPSU, but remove the controller from the equation.
With the machine off, unplug the power from the LaserBoard. You do not need power on the board for this test.
Remove the large white power connector from the board.
Work on the connector somewhere where it will not touch anything conducive.
Power the laser machine back on.
Set the pot / digital keys to a low power setting which is high enough to trigger the laser.
Use a short piece of wire to connect the contact metal of the Laser Fire (L) and Gnd (G) ports on the connector. This simulates completing the circuit.
Turn the machine power back off, allow the power to discharge before making any further wiring changes.
Is the laser firing when shorting the wires on the connector?