It would alert the authorities to the fact they might require emergency support, or might be going down.
Right - which is why it's on that list at all. But again, it is at the end of the list. A crew with an emergency that threatens the aircraft is generally going to deal with the emergency first, then talk to someone. It does no good calling for help if you hit the ocean at 650mph. A controller can't do anything about that; only the pilot can.
Take the alternative: if they detected a fire, do you think they'd just keep that information to themselves?
I'm a pilot. And I can tell you for sure that if my airplane was on fire, the very first thing I would do is try to put it out, then try to get the airplane on the ground as quickly as possible. If, in the interim, I had time/resources to declare an emergency, I would.
But let's take a very simple case. I take off, then at 1000 feet off the ground, the cockpit fills with smoke. First job - fly the airplane. Make sure it is trimmed, locate the nearest emergency landing area, turn towards it. Next try to put the fire out. Turn off the electrical master. If that fixes the problem, and the smoke starts to thin out, great. Keep the master switch off and land as quickly as possible. No opportunity for a radio call there.
If there's more time, and it was night (i.e. I will more likely need power to land safely) then I might turn everything off, turn the master back on and one by one try turning systems back on until I got the minimum set of instruments back on that I need. (primarily panel lights and electrically powered instruments like turn-and-bank.) That might get me the radio and transponder back, it might not. If the problem fried the voltage regulator, then I still wouldn't be able to talk to anyone.
We're at the point here where people are inferring that they went to 44K feet to put the fire out. I think they'd want to explain those air acrobatics to a controller somewhere.
They might want to explain - but again, that's going to be a lot lower on their list of priorities than "save the aircraft and passengers."
That would be one hell of an inferno, wouldn't it?
Yes, it would. It would be like the fire that brought down the Swissair flight. From the report:
"The investigation was unable to confirm if this arc was the "lead event" that ignited the flammable covering on MPET insulation blankets that quickly spread across other flammable materials.[1] The crew did not recognize that a fire had started and were not warned by instruments. Once they became aware of the fire, the uncertainty of the problem made it difficult to address. The rapid spread of the fire led to the failure of key display systems, and the crew were soon rendered unable to control the aircraft. Because he had no light by which to see his controls after the displays failed, the pilot was forced to steer the plane blindly; intentionally or not, the plane swerved off course and headed back out into the Atlantic. "
I really think they'd have called this in waaay before then - and before turning off communications. They turned off the transponder too. I don't know if a fire would have hit those systems first, but it seems like a lot of a coincidence.
Again, "shut everything down until the fire is out" might have been their plan.