> Who will pay when those people wind up in hospital, racking up bills of hundreds of thousands Euro, and there's no insurer to cover the cost?

Health insurance. I'm not aware of them not paying for obvious things like accidents, etc. They do avoid paying for things that are not strictly medically necessary, e.g. cosmetic surgery in many cases. But if you get a ride with someone and they get into an accident where you are injured, the question of hospital bills should not arise. At least not for the victim.

The victims insurance will for sure claim damages from the driver. As the driver most probably will not be able to pay the victims insurance has to cover the loss. This on the other hand means that the insurance needs to raise the prices for all their customers in order to cover for those kind of cases. To prevent that it is mandatory that all people are properly insured and pay their fair amount in respect to the risk - so a commercial driver has much higher insurance costs than a private driver.

  Health insurance.  

Ahh, exactly as I said: society at large pays for the profits of a private company.

And actually (at least in the case of Switzerland) you're wrong.

Health insurance does never pay for accidents, period.

In Germany health insurance does not cover medical procedures if they were caused by someone else (or at least they try to get their money back with lawsuits).

And the insurance of the driver wouldn't cover commercial activities, so he'd have to pay everything on his own.

Good luck finding a single driver for Uber then, because just one accident could bring a driver 300k$ into debt.