People on both sides of this issues are heavily guilty of "what if"s, there's no denying it. There's no possible way you could be prepared for every what if, especially not without sacrificing patient care to some degree. A gun isn't always the answer, especially if you have to shoot fast. Contrary to popular belief, most people can't hollywood a shot and take someone out the moment they draw the gun and with one bullet- people, the persistant fools, tend to die a little on the slow side when compared to hollywood deaths. Of course, all of us know that. Now, don't bother saying "but the aorta, but this vital organ, that vital organ, headshot, etc," because, like you, I know a bullet can kill you near instantly, but unless you've got some wild skills, you're not going to be able to nail a relatively small target like that at distance in a real scrap on purpose. I'm not knocking your skills, reader, I'm just saying.
Back on track, though, firearms in thickly urban environments present a problem nobody's considered here: bullets sometimes don't stop when they hit, and they certainly don't stop when they miss. We've all heard of real cases were innocent non-combatants were hit or even killed by a stray bullet. Guns are sloppy weapons, lacking in control, unless you're extremely precise in your method. You're massively more likely to accidentally shoot someone than you are to accidentally tase them (especially given that you have to hold the trigger down to keep the current flowing) or accidentally stab them. Let's be real here, when the fur's flying and so are the bullets, you're not going to be entirely too worried about missing- you're going to be focusing on destroying the threat before the threat destroys you, and you're going to miss unless they go out in a t-shirt made to look like a bullseye and stand perfectly still in the wide open for you while giving you plenty of time to get your aim dead perfect. Even if you do hit, that may not be the end of it. You might have to hit them multiple times to put an end to the threat.
Supposing you stop the threat, we also need to consider the ethics here. Do we turn right around and treat the man who just tried to kill us? Can we really go beyond ourselves to provide the same level of care we would to anyone else? How will this work with the job that Law Enforcement would have to do, which would invariably involve us in an investigation? How would our employers, our colleagues, and the public we serve respond?
A while back, well before this thread, I made a poll on another forum, asking whether or not people, uninvolved in EMS, would feel comfortable if we carried tasers after I explained why we might need to carry them. 90% of those who responded felt that it was largely unnecessary and that they found the thought of such a thing to be uncomfortable. To be honest with myself, I don't think I'd feel any more safe if I carried a firearm on duty.
Now, for most of the scenarios I can think of where we'd have a close encounter with a violent patient, a TASER would work wonderfully. It's safer, there's less complications, less tarnishing of the EMS name or social acceptance, and it works, else the LEOs wouldn't carry them. Sure, the barbs might not connect, and your gun might jam OR my partner might use his or her TASER when they see mine failed. The guy might be so jumped up on crack that the taser might only slow him down at best, but if it's the case that an electric discharge barely changes his muscle function, what makes you think he'll care about or get slowed down by getting shot?