Any operator for whom I have done ops engineering has had the special procedures discussed with and filed with the airports concerned. It is still necessary that the pilot calls it at the time .. but it seems silly not to provide the airport with the procedure if it is not going to be able to follow tracking which ATC might expect ...
Philosophically, I don't like the idea of having to tell ATC what is going in when the heat is turned up .. too much chance of the pilot's making a mistake in his story ... similarly for the ATC-er to mishear something ... doesn't seem to be any benefit to anyone in not providing the procedure to ATC in the first place. I am quite sure that many operators don't and that may be a consequence of specialisation .. ie the guy doing the work may not appreciate the problems faced by the other fellow ... at the end of the day, no simple solution to any of these problems.