The participants are in it for the profit, not for saving lives. The US health care misery is a prime example of what happens when you try to solve a situation through the market, which by definition has to treat human lives as an externality: People in the US die earlier than in any other first world country.
The causes of death might not be the healthcare system ,no? What if the majority of these deaths are caused by bad diets and stress? Would you blame the healthcare system for it?
You're right, one can't isolate one institution for topics as complex as these. I'm pretty sure it plays a role, though, as (ignoring fundamental differences in poverty rates, workplace regulations etc.) many countries employ the healthcare system to inform the population about healthy eating and offer incentives and free courses for coping with stress. Thats the beauty of a single payer system: Beyond ethics, the healthcare system has an intrinsic interest to prevent the preventable, as that is almost universally cheaper than bearing the cost of circulatory disorders later.
There is no data to suggest that this will happen. Furthermore, if self-driving cars do take off, they will continue to be outnumbered by normal cars for a very long time. It's not even clear that they will or should be the majority.
Now if instead the road deaths stay the same or go up, then naturally that should should also spell the death of av technology.