To not be able to transmit in an emergency situation is a serious flaw that needs to be corrected.
If the Air France pilots were so maxed out by the situation they were in (and this is just supposition) then what value would have come from a radio transmission from them? They probably wouldn't have had the physical and mental bandwidth to communicate any useful information.
In a sudden and extreme situation it it is likely they would have been trying to disambiguate multiple readings, warnings and symptoms.
As it stands, we don't know whether they attempted any transmissions on HF.