OK, but this supervisor will need flawless communications, a very good knowledge of the aircraft's route network, the aircraft's capabilities and limitations( in fact he/she might need an old fashioned type rating) and I'd suggest to avoid the odd unnecessary diversion the supervisor needs a feel for what is really going on in the cabin and the state of the patient, the sort of thing you only get from face to face communication with the cabin crew, and I don't mean a video link.... sound's like you still need a decision maker on board to me.
I think not. Anything from a big red "oh !!!!" button to a keyboard and monitor to send the nature of the emergency, could have quick buttons... etc. Why the heck do you think we have developed an ADS-B network and are using a CPDLC system???