Why does LM317 burnout when output is shorted in this case?
When you short the output, Q1 heavily conducts and basically connects pin 1 (ADJ) directly to 0V. Between Vout and ADJ internally is a 6V zener diode and a 50 ohm resistor: -



It's very likely that the zener diode will fail short circuit (most of them do on over-current) rendering the device dead.

If it can be tolerated a 1k resistor in series with the ADJ pin will probably save it. The 50 uA ADJ pin current (normal operation) will cause a 50 mV error (times the standard R1/R2 feedback ratio) in the output voltage so there is a somewhat fluffy downside potentially.