Does anyone know about an incident in which an emergency call should have been given, but where the crew was reluctant to do so, or maybe the otherway around, where they did, but that it was a bit overdone?
As far as ATC are concerned the pilot is the only one who is qualified to make the decision about making the call. We would never treat a PAN or MAYDAY as anything other than an aircraft in emergency therefore it has NO1 priority over all other aircraft. At that point the pilot gets what they request. Direct routes as soon as safe (we would even get danger areas closed if possible), priority approach, clear area to dump fuel etc.
ATC can only give as good a service as is requested. An occasional update and sometimes non technical explanation of the problem is usually a great idea to clarify the position and then we might be able to give more useful information. i.e. if you have an engine shut down and don't want a cross wind from one direction, what are the runways in use and wind strengths and directions at the nearest suitable airports ?
I did once have a large military a/c which requested a direct route to a military airfield. Several minutes after that request he stated the condition was under control and the PAN cancelled. Unfortunately the original PAN call had been so brief that it had partly overlapped another R/T call and I had not heard it !! So if in doubt say it again and always use standard phraseology.