Oceancrosser is raising a very valid point here, this was in the US!! How about if it was in France, Spain or China?!
Possible it would have been smoother. Possible the ATC would have understood "emergency, smoke in the cockpit, roll the trucks". In fact I cannot recall that the sentence "emergency, smoke in the cockpit" was once not understood by the French, for example, ATC.
So if the ICAO language is english and if basic simple sentences are not understood in the US, maybe that's time to change language
A good old MAYDAY MAYDAY MAYDAY would have worked, funny when you know that it directly comes from the french M'AIDER M'AIDER M'AIDER wich means in french HELP HELP HELP.
To be understood, speak french my friend!
The Mayday callsign was originated in 1923 by Frederick Stanley Mockford (1897–1962).[4] A senior radio officer at Croydon Airport in London, Mockford was asked to think of a word that would indicate distress and would easily be understood by all pilots and ground staff in an emergency. Since much of the traffic at the time was between Croydon and Le Bourget Airport in Paris, he proposed the word "Mayday" from the French m’aider. "Venez m'aider" means "come help me."
And honestly, they are very lucky on that one, this mistake could have cost a lot in human life if the fire had been more serious.