I have found it alarming that at least 2 ATO’s and a hand full of instructors I have met refuse to teach trouble checks after having an engine failure. Checks like these MAY save “Possible” situations like this (if fuel system orientated). Also the number of pilots that are taught to flick the fuel pump on as a first response to any engine problem. How ever if fuel starvation is the cause turning the fuel pump on will just cause the ingestion of air into the system and take longer for the engine to come back to life when the tanks are changed causing loss of valuable altitude.