I think there is a big difference between a long haul and a continental (US/EUR) flight, where you can be on the ground before you even had the time to consult a ground based service.
Had a person pass away on a long haul, initiated diversion but in the time before TOD already consulted 2 doctors on board and one on the ground. We continued to destination where it took more then half an hour before the aircraft was cleared and the first passengers were allowed to deplane, while the corpse remained on board. It looked like a crime scene out of a movie, including loads of police and a fully zipped up coroner.
What really is an issue that it made quite an impact on the crew and I wonder whether I would really have wanted to continue to our destination if it had been an enroute diversion.
(we also have a body bag btw)