Why does the fire department respond to a medical emergency?
It makes sense to staff FDs with EMTs and give them responsibility for responding to all manner of emergencies (medical included), since they are already the first responders.
That being said, I live in a rural state where this makes sense. I cannot imagine the crowded streets of SF are good for this.
It's all about level of care. Overkill for a twisted ankle sure but crucial for a cardiac or other life threatening event.