I think it's more likely that your MOSFET is blown. They are very sensitive to static electricity. When they blow, the gate can get shorted to the drain or the source. If shorted to the source, that would explain your S9013 getting really hot.An update on the original board.
I just went over the board again with the same results as last time. Thought I would post them in hopes someone could give me a hint of how to fix it.
Here is what I found before and just found again now.
The fuse was blown. I jumped in a similar .5 amp fuse. The fuse did not blow but the light did not work either. I did notice that there was one small transistor that gets very hot. It is an S9013 and it's collector is connected to the gate of a F9540N power Mosfet. I believe this is the MOSFet that switches the solar panel off after the battery is charged. The solar panel is connected via a resistor to the source of the mosfet.
And if this circuit is malfunctioning, it might make sense that the light does not light because the board senses the voltage on the panel. It turns the the light on when the voltage is zero (or whatever low output they picked) from the panel.
Could that S9013 be shorted?