http://www.nytimes.com/1987/01/31/style/private-ambulances-w... (yes, over 30 years old, but most of it still holds true)
There's no theoretical reason why Uber couldn't get into the driving side of the business, with human drivers. Their future is in autonomous however, which will never be the ambulance business (it will never be autonomous, ambulance drivers frequently have to break standard driving laws in emergencies, drive at high speeds, drive around vehicles, etc).
You'll still need warm bodies in the back for life support, but I see no reason that the human in the front seat can't be replaced by sufficiently advanced technology.
Hell, in this ideal future where most or all of the rest of traffic is also autonomous, you don't even need to avoid congestion -- you just "phone ahead" along the route to tell any cars in the way to, well, get out of the way. As long as you don't have standstill traffic (which might actually be something you can eliminate as a general rule with a fully-autonomous, networked fleet), you're fine.
And because they have to perform emergency medical treatment, right?!
Are we talking about different things - a mini hospital on wheels staffed by highly trained paramedics, and glorified taxis for the elderly/non-critical?
If it weren't for the need to drive in a "non-standard" manner, you could likely save on costs by having an autonomous vehicle staffed by an EMT. The EMT can't also drive the vehicle, after all.
None of these are reasons for ambulances to not be autonomous. If you accept that most other vehicles will be fully autonomous, ambulances do not require anything further extraordinary to also be autonomous. The vehicles on the road would likely be networked, at least within a nexus of proximity, and move out of the way of the ambulance so it could speed past them.
NB: I’m not saying ambulances will be autonomous. I’m just saying that if everything else is, the ambulances have no significant (technical) obstacle for also being autonomous either. A significant theoretical advantage of fully autonomous vehicles is not simply that the modal vehicle is safer than a human, but that a networked and autonomous fleet is hyper-efficient and hyper-aware as a hive mind.
Shoot, it'd be safer even if the ambulances weren't autonomous - because the cars would be reacting in a predictable manner.
It’s a real mix though... paid fire dapartments, volunteers, and private operators all respond to emergencies.