I wish I had known about this thread before now. This is where I came in and I wanted to comment on these posts before reading the whole thing and throwing in my $.02.
Diversions. Medical diversions in particular are tricky things. First a short story: Going to Beijing via a polar route and through eastern Russia. Our divert fields at this particular point in time are Fairbanks Alaska, and I don't remember if it was Magadan or Petropavlovsk and finally Beijing. We were literally at the equal time point between Fairbanks and Beijing and had a passenger in pretty serious trouble. Magadan or Petro were closer, no doubt. We have a medical person on board that has an opinion but says she cannot make a real diagnosis without lab work. We have the capability to hook up an on board medical person with a medical service we use (I call it Doc in a Box). I don't want to get into too much detail on that. We hook them up on SatCom and they discuss possible diagnoses and what our potential divert situations are. Now if I am on fire, I am going to land in Eastern Russia. For medical reasons... probably not because we might very well have better equipment on the airplane. The other part of this story is the fact that the cockpit crew has to coordinate the diversion process with ATC, and while not that difficult, even with the Russians, buy the time we would have coordinated it with ATC, and our dispatcher (who tracks every facet of our flights) we would have been past the ETP anyway and going back to Fairbanks would have taken longer. We proceeded on to PEK which was still 4.5 hours away.
A little epilogue to this particular story: When I was going back to my rest break I went back and talked to both the passenger and the doctor. The passenger was a very nice man who was obviously in a lot of pain, but he looked at me and mouthed that he was sorry for all the trouble. I felt terrible, but there was nothing else I could do for him at that particular point in time other than to tell him not to worry about that and that getting him quickly and safely to the best care we could was a top priority. We were met by an ambulance and he was taken to a hospital. As I saw him getting off, he seemed much better, but still not feeling the best.
So... from MY perspective in the cockpit. I want to go wherever I can get the passenger the best care as soon possible but if I am in the middle of the Atlantic Ocean or over Eastern Russia it is not always close. It might be 3 or even more hours away. So you are probably asking what about over Nebraska. A lot of times, the FAs will call for a doctor before they ever let us know up front about it. And 'doctors' can be any health professional from EMTs to OB nurses to neurosurgeons (I have had all of those). The Doc in the Box helps us to make not only the decision to divert, but WHERE to divert to, depending on what help is needed. That is information he has at his fingertips. For example if it is a heart issue, we might want to go to a city that has a good heart center as opposed to finding the first piece of concrete available. Oh, and cost never enters the equation.
Sorry for the length. I understand tldr if that applies.