First of all, if I call you onto the field of play, DO NOT try to act like he/she is YOUR patient. Guess what? When I am hosting an event, my team physician and I ultimately have the responsibility for every player, coach, official, etc on that field of play.
With regards to exertional heat stroke (EHS) it is crucial that cooling is started IMMEDIATELY! If your protocols specify transport first in an EHS case, you need to be talking to your supervisors. This is one of probably very few cases where EMS transport must wait. For football in the heat, I keep cold towels, lots of ice, and a rectal thermometer on the sideline. I keep my whirlpool inside filled up. If we have a kid go down and suspect EHS, the cold towels will go on while we transport into the ATR to get him into the tub. Once we have the core temperature down, then we will transport via EMS.
I would recommend you trying to set up training sessions with your local athletic trainers. We may not deal with the same emergencies EMS does on a daily basis, but we can and should be your biggest allies when it comes to sports emergencies. We are trained to handle the equipment commonly found in sports.
Technically and legally, I operate under the direction of a physician just like EMS does. But realistically I call the shots on the field. That is my job. And when emergencies strike and we initiate EMS, we must work as a team for the betterment of the patient.
I had an incident earlier this year which necessitated EMS activation. From the time I initially arrived at the patient on the field, I never took my hands off the kids head/neck until he was strapped to the backboard. At that point, he is "your patient."