Wrt your first inst. David, given the stop would be in mid/north Africa I would prefer the pain than take a risk in African hospitals.
If it was in the US then a stop would be preferred.
Your final point is interesting, maybe you could get some element of reimbursement from the pax insurance company but if one were to accept the principle that the venue of the emergency should chip in then it opens a much broader discussion (especially as many Doctors have opted out of late call-outs).
Interestingly Virgin sourced many, if not the majority, of their cabin crew from the NHS when they first started up. They knew the service, caring and people orientation of nurses would make ideal Cabin Crew – just offer them a bit more and recruitment solved. Plus you get the bonus of medically trained people on board.
Edit – The type of aircraft is not disclosed, however I thought most LH aircraft had crew bunks for rest periods. Were there any bunks? Was it too painful to manoeuvre the patient?