Mayday
There is alot here on this thread and not sure if I missed it being mentioned but one advantage in issuing a mayday would not only be to advise ATC of current situation but also to alert other aircraft on frequency of the requirement for the comms to be kept to a minimum so assitance can be given.
p.s. I do understand from the CVR that a mayday was issued but communications interrupted it - perhaps this is where technology could be used to improve dialogue.