No, it has to do with Operational Priority.
An aircraft with a sick passenger on board, even one nearing death, does not get unlimited operational priority (meaning, right of way over all other traffic).
So in your Cairo example, even if the pilot had said PAN PAN or even MAYDAY, the nature of the emergency does not rise to the level where the ATC must drop everything else to handle the flight.
See for example FAA 7110.65V, specifically 2-1-4 (a) and (b).