Not a 'Mayday button' at all!
Dialing up the 7700 squawk on a transponder at a time when the crew have their hands full with an emergency at what I presume was relatively low altitude is going to be very low on the list of priorities. In a critical situation with little time available probably the only viable option would be to key the mic button under the thumb and make a 'Mayday' call on whatever frequency was currently selected on the radio. But a 'Mayday' call might not have crossed their minds for even that small action would have demanded mental capacity to be diverted from the real requirement of handling the situation.