I acknowledge that looking like LEOs hinders our ability to do our jobs, and that carrying a weapon helps us look like LEOs. Additionally, I know that to carry any weapons requires extensive training, and legally, could be ugly if the weapons were ever used, but I believe that first and foremost we have the right to be able to defend ourselves whenever necessary, and do what is necessary to go home at the end of the day. It is important to keep the image of EMS positive, but more important to keep EMS as safe as possible. Carrying weapons for self defense doesn't necessarily have to be obvious. Ever visited the White House and seen a Secret Service agent? If you didn't know what their job was would you know they were armed? It is possible to preserve the image of a benevolent organization while still allowing people to protect themselves with more than their pocket knife and an O2 tank (which would really hurt).
A good EMT/medic will evaluate every situation to try and keep an exit open, and people in front of them, but EMS is dynamic, and as such, its not always possible to have these preferable conditions. What if you got backed into a corner by a violent patient? You may have a knife of some sort, blunt trauma shears, an O2 tank, but only if its in reach, and maybe a maglite if its a night call. All fine and good if said violent patient is unarmed, but have you ever been in a knife fight or gone up against a person with a baseball bat? I would want more than just the standard EMS equipment.
The situation presented is not a wild idea. The police often leave a "secured" scene to handle other things, leaving us to fend for ourselves. Additionally, we respond to many calls without police. Many who have worked in EMS know that we can be dispatched to "difficulty breathing" only to find that they are, in fact, having trouble breathing because a bullet or knife punctured their lung. When was the last time police responded to "trouble breathing?" I'm all for running away as fast as I can, but it may not always be an option we have, and kevlar, if its even worn, only does so much. At a minimum we should be able to get trained in the use of, and be allowed to carry OC spray or a taser, and would not object to sidearms. When I go out I always have my armor on, and in addition to a pocket knife and a Leatherman, I keep a second knife tucked in my waistband by my back pocket, not because I expect a dangerous situation, but because I want to be prepared for one.
