Ambulance and fire truck driver here. There's no good reason for emergency vehicles to ever go much faster than the speed limit, and we would experience life-changing amounts of personal liability if our driving got someone hurt.
While it's sometimes important to get a patient to the hospital as quickly as possible, that's less frequent than you might think, and it's always more important to get them there in one piece.
In addition our vehicles are heavy and they don't stop quickly, so physics is another good reason for us not to speed.
Police cars might be another story but my personal opinion is that speeding police cars probably don't create a net benefit for public safety either.