Captaincy is a valid point.
The reason why I think a “discussion” about whether to call a pan or mayday was the right call was because there was no immediate danger or need for urgent assistance at that moment. There was no reason to think the engine would not restart as it had been intentionally shut down. I was happy to do neither and as it turned out so was the examiner. Captaincy in whatever situation, even in a test, is about using the resources at your disposal in the best way. Personally I would happily consult with any pilot in the seat next to me if I could. In the same way, whilst you may well jest about getting on the ‘phone, I would happily call ops from the air if I thought they might be able to assist with an issue once I was happy the aircraft was stable.
If the engine has failed, or in this case, had refused to restart, on balance I want the urgent attention of AT. If I was enroute and was talking to a unit other than that at my diversion a pan might be sufficient, but it is still going to me a mayday at my diversion, so again, on balance why bother with the pan.
The fact is a single engine approach is a mayday situation imo .. .. ..
.. .. .. unless, that is, you know of so other way of getting back on the ground?