When I'm running, I'm not able to get very far away. No way will I be able to leave the area covered by our local hospital. I normally run in the town or in a park, where people are passing by all the time. And should they bring me to the hospital unconscious or something, I'm pretty sure, someone will recognize me. I have been a doctor there for almost 25 years. But of course, I may change my habit and start bringing a phone. I have actually been thinking about the safety aspect of it, as you mention. I always bring a phone on the bike. I just think, that when running in the summer wearing only shorts and a tee-shirt, a phone is rather unpleasant to haul along.
With bike, I of course mean a road racing bicycle with 20 speed, and not one of those with an engine