Is it possible for a circuit to heat up enough to bridge solder?
I would suggest that the solder bridge has always been there and that the LM317 was never doing its job as a constant current supply. The current sense resistor was shorted out by that solder bridge.  Your LEDs will have been getting overdriven by excessive current all the time and have eventually failed.
It should be obvious from looking at the resistor connected to those pads if it has overheated, it will look burnt.  I think you will find that it looks ok.
I think once you have removed the solder bridge the circuit will work again.  However,  your LEDs will be broken in some way.
Since the LM317 is configured as a current regulator in this circuit, either off load or connected to a higher resistance faulty load, it’s voltage output will be higher than expected.   You could measure the current supplied to your good LEDs by the good board and check that your suspect board supplies the same current to the good LEDs.