I agree! Anything that has to do with a patient's care is totally tied to patient extrication and rescue. Let's follow this concept!
So, an accident that needs patients extricated is DEF tied to patient care. This will require hydraulics which need a generator. We'll need a pump and water tank for the charged handline as required by NFPA, as well as proper extrication PPE, and possibly struts/jacks/rams/airbags/cribbing. This may also require set of irons, and cutter for the batteries. Also, a saw or two couldn't hurt, with extra blades and batteries/fuel. Also, tarps. If the vehicle is engulfed when we arrive, we would need airpacks to extinguish it, as required by both NFPA and OSHA. I would want to extinguish it myself to ensure I don't cause steam burns to my patient. If it's dark out, we'll need a light tower.... or four.
BUT WAIT! Little Timmy just fell down the well and is injured! Well now we'll need properly trained people, and at a minimum a few hundred feet of life-safety rope, a tripod, harnesses, rack, and other proper tech-rescue equipment and training. If this well is far from a roadway or down a ravine, we could also use a stokes basket and more rope and manpower, all rescue- and medical-related, of course.
But now, a tank is leaking some fertilizer and people are sick! Clearly only rescue-trained Paramedics can save they day, and we need to get to our patients. I now require Level A suits to go with my SCBAs, along with proper decontamination equipment and that watertank/pump/hose combination again. I'll retrieve and decon these people myself to ENSURE proper patient care.
What's that, a person swept away from a ditch during a storm? Well good thing we have our RIB and lifejacks (floaties?) along with cold-water-immersion suits, because when they patient gets onboard I'll need to monitor them for hypothermia, initiate rewarming, and possibly even perform CPR or intubation immediately.
And then I got to thinking, what if there's a workplace shooting and people are wounded inside? There's been multiple instances of workplace or school violence where deaths occur because rapid care wasn't given to victims. And I need to save them, so my rescue team will need ballistic helmets and protective vests, possibly shields. But now to protect ourselves and our patients, we'll all need to be armed. No sense going about this half-arsed.
....On second thought, maybe a tank, so we can just DRIVE ON IN THERE TO EXTRACT THEM OURSELVES.

h34r:
tl;dr - I get it, EMS is still the redheaded step-child that is trying to make a name for itself. Fire is the bratty older brother who always gets the cool stuff and gets what it wants. The fact remains, we are three pillars of public service that cannot stand alone without the other two. We should each stick to what we do best. There's no denying, our common goal is proper service to our patient. My job is patient care, and if I feel there's an issue with my patient or extrication, I'll speak to the person doing it. Otherwise, I don't care who cuts the damn car, as long as my patient gets out. I am VERY EMS-oriented, and stress proper patient care at all times. But turning over anything related to patient treatment to us? It's a bit much.