Might not be a bad idea, Hugh
As an interesting side note, one of the clients at my shop (before the wreck) was SWAT team triage leader with combat medic experience who oversees the three SWAT team triage members and is a primary ER triage technician at one of the two local hospitals.
Turns out that when he heard the 911 call came through dispatch regarding my wreck, he came and took over my care at the crash site from the paramedics and remained with the team until I was stabilized and transferred from ER to the Intensive care ward.
I have high regards for EMTs, but nothing beats a combat medic - so I was in good hands.