Your LM317K's are fake.

Buy a switching regulator like the EZSBC as it is guaranteed to be short circuit protected and won't blow up if you have some board fault.

There are just too many FAKE LM317K around to trust ANY of them these days. Don't waste any more time of money trying them.

Another tip - don't shotgun parts for a repair, it rarely ends well and often introduces additional faults. Troubleshoot LOGICALLY and replace ONLY what is actually causing the fault. After fixing the fault, it is sometimes prudent to replace known troublesome caps etc but please don't do this as if it is normal repair procedure - IT ISN'T!

I would ditch that LM317 altogether and replace it with the fixed 5V switching regulator EZSBC.

Only mods I can see you would need to do are ground the pin that currently goes to the resistor dividing network (R1 & R2) and rip out those two resistors.

I can't see any reason NOT to make this small mod?

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There won't be any markings on the die but its size will quickly give the game away.

The wires might struggle at 3A?

You would have to chop a known 'real' one to know for certain. The dies can be a lot smaller than you might think for some parts.

Nice job! Notice there is a LOT more metal that the die sits on than in the fake. Each of those connecting wires could easily handle 5+ amps so they are certainly adequate for this part.

Most likely something super common and cheap like a 2N3055?

If you have one of those cheapo component testers it might find it?