Batteries powering off immediately after power on when inserted into the bird would indicate a short somewhere as at least the newer batteries have a short circuit/over current detection that will power off the battery. I would agree to start with the ESC boards that drive the motors and visually look for anything that looks to have been overheated, burnt, melted, or does not look like the other components on other boards, especially on the suspect motor arm as the motor could have stopped under power and burnt out a component on the ESC board before the power was shut down. I would also try if possible to inspect the motor windings that can be seen with a strong light source and and magnifying glass looking for anything that may indicate winding damage (burnt spots, impact damage, cuts in varnish or foreign matter in the motor. A damaged/shorted winding or burnt winding could cause an over current shutdown.