Even if the healthcare system eats the cost (as it would in my country), it's not supposed to. That's akin to entering a health insurance contract and then demanding the insurer pays for the theft of your iPhone.
Road traffic is insured by car insurers. The Uber driver doesn't have any, the car insurer therefore doesn't pay, so someone else pays. It doesn't matter if you rationalize it with "oh, that someone else is also some insurer". It is someone not involved at all. Ergo an externality.
I am yet to meet a single person complaining that this is bad. There is no increase in road accidents. The service is great. The drivers are very polite and are always on time. One must wonder, how can you be so blind to an obvious example of free market at work?
When there are no pax, many insurance providers DO cover it under personal insurance. Some (like geico) don't, but I would wager that the majority of Uber are insured every second of the day.