Having firefighters trained and certified as medics is questionable but not as out there as combining all three realms of public safety - firefighting, EMS, and local law enforcement - into one discipline. However, I do see the merit is some overlap with EMS duties with firefighters and police...
That's a lot of training for an individual. Paramedic training alone typically ranges from 18 months to two years. I personally do not believe that someone who is cross-trained and certified as a cop/ff/and medic or EMT would be as efficient as an individual who is solely certified as a police...