However, in these kind of situations, I don't think it's in the patient's best interests to avoid diversion. The algorithm should be: 1) sick vs not sick - this person was clearly sick (when an ER docs say someone is 'sick', they usually mean there is a non-trivial probability that they could die in the next 24 hours) 2) Could a delay in critical care treatment lead to a worse outcome for the patient? If so, I would argue that you have to advise diversion to the nearest airport with the required standard of medical care.
That means that it might be ok to advise taking one hour to get to a major city, rather than taking 10 minutes to land at a rural airfield with a tiny hospital nearby - that's a judgement call. The only reason not to divert that I can think of is the cost to the airline, and passenger inconvenience: both of those looks like really bad reasons if the guys gets worse again and dies on the plane, when they might easily have been saved if they had diverted to a nearer airport + hospital.
I agree with another comment regarding the difficulty of IV's, particularly in shocked patient. Ideally, the airline kits would stock intra-osseous needles (needles with a screw that are screwed into the bone of the shin or upper arm using an electric dril). It sounds brutal,but is probably not much more painful than an IV, and takes seconds to do with training. The crew could be trained to use them as part of their first-aid training - the training takes less than an hour, and it would probably make more difference to patient outcome than having adrenaline on board would.
I totally agree about speaking to the pilot in person when you have a critically ill patient - they know (or can figure out) flight times to the various possible diversion airports, you (probably) know better what kind of care the patient needs, and the chances of finding that kind of care in a given city.