The point you're missing is that they don't have to get to an emergency room since an ambulance, unlike an autonomous vehicle, has trained medical personnel on board that can start administering procedures immediately upon arrival.
Also, FYI, despite what you may think, ambulances DO have to follow the speed limit.
It varies by jurisdiction, but no, they don't. At least in California, they can legally exceed the speed limit:
(b) Speed Restriction. In no event shall an ambulance driver exceed a speed of 15 miles per hour while disobeying any official traffic control stop sign or stop signal.
So that means that while going through, say, a red light, not only can they legally go through it, but then can, at most, exceed the speed limit by 15 MPH while doing it.
While NOT "disobeying any official traffic control stop sign or stop signal," they can generally exceed the speed limit even more (but, again, it varies by jurisdiction).
Here is the relevant law for Hollywood, FL:
No ambulance shall exceed the speed of 40 miles per hour except in cases of extreme emergency.
Which means they can exceed the speed limit by 39 MPH for "non-extreme" emergencies.
In New York State:
1104 Authorized Emergency Vehicles - (b) The driver of an authorized emergency vehicle may: 3. Exceed the maximum speed limits so long as he does not endanger life or
property;
No specific maximum speed is mentioned.