The medical director of my paramedic program used to work for a hospital that provided medical control for a number of major airlines (generally speaking, even when they ask for a physician on a plane, they're also calling the airline's 'on call' doctor as well).

His scariest stories weren't due to the criticality of the patient, but rather the incompetence of the provider on board the plane (to the point that he, on several different occasions, had to instruct the flight crew to 'please keep him away from the patient for the rest of the flight'). Just because someone is a 'doctor' doesn't necessarily mean they're well equipped to handle a medical emergency (note: the physician interviewed here managed the patient perfectly appropriately).

It's possible he was just pandering to the room of soon-to-be paramedics at the time, but he said he generally instructed the flight crew to ask for a paramedic before a physician, simply because they're generally more comfortable working in the 'austere' medical setting found on a plane.