That's not a really big limb: It's not as chaotic as it looks to the untrained bystanders and the armchair medics. The point of the on-scene activity is to get people to "definitive" care as quickly as possible.
According to my trainers in emergency care, standard mass casualty protocol is that unless there is a shortage of first responder personnel, transport or an access problem, you move as many of the wounded as possible off the scene as fast as you can evaluate and stabilize them. If there are shortages, you adjust the plan to save as many lives as possible. (yes, first responders have to "play god" sometimes)
You stop minor bleeding, check for head trauma and level or consciousness, maybe splint injured limbs, but the walking wounded are often on their way to an ER while the severe injuries may be still getting stabilized or extricated.
Triage can take place in ambulances ... you load several lightly wounded people in one vehicle and take them to a smaller hospital or an off-scene treatment spot because that's less of a load on the trauma centers.
So, the EMT/driver lets dispatch know what she's hauling and is told which hospital to go to. That hospital is told what's coming.
BTW: Hospitals are automatically notified to be on mass casualty standby if anything goes kaboom or was likely to have produced unusual numbers of injuries. It's not a big deal - you just go to alert and stand down if it wasn't needed. And if it's for real, you do what you are trained to do.