Most people aren't capable of driving as fast as a trained emergency responder, much less driving like a psycho in a panic because they're dealing with a medical emergency and are completely emotionally unprepared for it.
I've been involved in an situation that resulted in a fatality and trying to rescue the victim and for an untrained responder your ability to think and process gets chopped down to around 25% of what it is normally. That is probably worse if the victim is someone you know. Now you want to argue that those people should be driving 100 mph through the streets blowing through red lights and whatever at their own discretion like that solves problems.
People need to call 911 and get the ambulance, which can stabilize the patient, to come to them. They can run code red and are trained in how to run red lights and get around traffic safely. They do it for a living and they're emotionally detached from the emergency.
Not to mention that the tires on the average vehicle shouldn't be driven that fast and they're probably nearly bald due to lack of maintenance.