Rodimus, you're not necessarily wrong, but what you're saying is not entirely correct.
The matter of who controls the scene, police, EMS or FD is a matter of contention between the aforementioned agencies and the laws, regulations and precedence can vary greatly between states/jurisdictions. This becomes even more complicated when when state and federal agencies become involved in say, mass emergencies.
There was a highly publicized recent case where video was taken of a trooper arresting a fire chief when the fire chief refused to move an engine he had set up to block. IIRC, the ultimate result was that the trooper was found in the wrong, so I'm guessing in that jurisdiction, unless a crime is being committed, the precedence has been set that the FD/EMS, not the police/LE controls the scene.
I haven't seen the video (limited bandwidth) but was the ambo involved a government emergency response vehicle or was it a private ambulance?
And BTW, first responder first aid training as is most commonly taught to police can be covered in a single day or weekend. To qualify as a paramedic (EMT-P) is at least a year, usually two, of full time instruction/practicals. The two are not even remotely close to being the same thing. That said, some states don't require a Paramedic credentials to drive an ambulance and a EMT-A or B will suffice.