There is the "tyranny of distance" as it is referred to in combat medicine, where the tactical situation prolongs extraction of a patient to definitive care. I have seen tactical situations where an ambulance was parked 100 yards away but took 30 minutes or more to reach the patient because of safety concerns and volatility of a situation. For extracting the patient, who will do that when the officers on the SWAT teams are engaged in a search for the subject or explosive devices? That's not a setting the average street medic is trained to be in. TECC training and the Rescue Task Force concept gets us closer to where we need to be, but this trauma care at the point of wounding is a small portion of what the SWAT medic does.
Many SWAT teams, mine included, do foot patrols to locate subjects in wooded areas that are a significant distance from a road or other route of extraction. Those medics have to carry not only their tactical protective gear (body armor, helmet, gloves, gas mask) but also water, snacks, and medical supplies. If you look at the average "first in" bag on an ambulance, it is not in any way designed for being carried a distance over rough terrain for a long period of time. So they need to have a kit that is specially designed with that in mind (it's more than throwing stuff together. Much of the kit has to be repackaged altogether, and interesting choices made to reduce size and cube.) And that whole time you are advancing on a subject that may be lying in wait for you. The average street medic has probably not been trained in tracking, terrain selection, and counter-ambush techniques. And then there is the prolonged field care during extraction from that scene. Carrying a grown man over rough terrain takes a lot of time, a lot of people, and cannot be done easily.
The SWAT medic is also responsible for team health, preventive medicine, minor injury care, sick call, medical contingency planning, and analysis of medical resources. No mission goes out the door without a medical plan and the SWAT medic is the one to create it. Appropriate hospitals for various issues, casualty collection points (primary and secondary) unconventional means of carrying patients there from and possibly through a hostile environment, conventional prehospital medical systems (EMS, flight) and their capabilities. Commo plans are an achilles heel at all times and need to be vetted, but no cop knows how to get hold or a hospital, so that falls to the medic.
A team that has a mishap grilling food for the group may find itself completely combat ineffective for the mission if there is a case of food poisoning. Someone has to think about that, and someone has to treat it to keep guys on the line. Something as simple as a dental issue can be a distraction for someone at the wrong time. A corneal abrasion, properly treated, is the difference between the operator finishing that mission or not.
SWAT medics fill many responsibilities other "take care of the guys that's shot."