Next time this happens, don't let the circle happen, start shouting orders at people. Become a manager on the spot. Tell them to do anything. Something as insane as telling them to do jumping jacks will work.
I've been in that situation before, I've been the first responder on site and while I have limited medical knowledge I knew enough to check pulses and verify that people were "okay", as soon as the second and third people showed up I started yelling orders at them. For some reason this causes them to do without asking "why" whatever you are telling them to do. Call 911, block off the road, turn on hazards, warn people coming down the roads, get a blanket, get water, I need a knife to cut the seatbelt.
Shout status updates to the person on the 911 call.
In my case a vehicle had caught on fire and I wanted to get everyone away as soon as possible. I started yelling at people to get out of the way at least a 100 ft and get down, I didn't want to risk an explosion (yes, I am aware of how unlikely it is, and it didn't happen this time) but there was a fire in the engine compartment. At that point a doctor had stopped, I was yelling at him to get away when he mentioned he was a doctor, I told him to help me get the person out of the vehicle that was on fire. Two other strong volunteers had made their way over to the other vehicle and were pulling the three people out of that vehicle (which had fared much better).
To this day I still don't remember faces, I don't remember much of what I did or how I did it, but I do remember a police officer telling me that I should consider myself a hero because I did what I should have done. He told me how the other people told him that I was yelling orders like a mad man and that I seemed to have the scene under full control. In reality I have no idea what went through my head, I have no idea how I was able to do what I did and what my thinking was like at the time. I had heard of the "By Stander" effect you mentioned before this, and I had been told what to do, but I have had no training at all.
To this day I still have slight PTSD. I have had terrible dreams about what I do remember, the person on the passenger side of the car which was on fire whose head was so smashed in from hitting the front dashboard (no seatbelt, no airbags) that it was completely missing, blood everywhere (blood is also surprisingly slippery ...) and trying to make sure everyone was safe that was showing up to "help".