voltage regulator

Using a multimeter on continuity test may lead to uncertain results. The continuity threshold is typically 40 Ohms. There is a diode check on some meters, it measures the forward drop of diodes if leads are connected to forward bias juntion. This is mentioned because mosfets have an intrinsic diode, and a flyback diode is in the circuit to shunt the field, protecting the transistor from inductive kick.

Transistors are rare to fail, but the most likely failure is shorted. To fail open requires huge current to explode the die. Since field resistance limits the current to about 5A, that may not happen unless the field is shorted, or 12V applied without field.

A good way to damage a regulator is intermittent ground or field connection. The field has significant inductance, when opened the inductive kick results in hundreds of volts. The volts are applied in an abnormal manner, outside of normal operating and protection means. Yes some might survive.

Here is good information on regulator circuit design, page 7 is most interesting.