You can't sue Uber if a driver crashes, because drivers are independent contractors. You can sue the driver, but he has no money and can just dodge the judgment in bankruptcy.
There's two differences there: 1) those suits are the result of special laws ("dram shop laws") that extend liability to bars; 2) liability in those cases arises when the bar did something negligent (serve someone who appeared to be drunk, serve a minor, etc).
Absent a special law, you can only sue a business for the negligence of its employees under the theory of respondeat superior: http://en.wikipedia.org/wiki/Respondeat_superior. The essence of the theory is that businesses should be liable for the risks created by their employees acting within the scope of their employment.
However, respondeat superior does not apply when the tortfeasor (the person who acted negligently) is an independent contractor. When it comes to employees, a business can reduce incidences of negligence through training, setting procedures, etc. They exert a great deal of control over how the employee does his work. That's not true in the case of independent contractors, like Uber drivers.
There is the basic principle that anyone can be sued for the consequences of their actions. Uber arranged that drive. They didn't check for insurance status. That could be cause enough. Plus between you and the driver there exists no business relationship. So you can argue that there's nothing to sue about.
Besides who would you sue ? "The driver" doesn't work too wel. You'd sue Uber to get the name of the driver in the first place.
This go-between-suit is pretty common in legal cases. What should happen is this. Judge agrees Uber owes your medical bills/restitution/whatever applies. Judge also agrees that Uber can get this amount from their driver. Then Uber has to cough up the money, but if they ask for delay in payment until their suit of the driver is finished, 99% certain they'll get that. Then you get the choice of getting the money from Uber, or joining Uber's suit against the driver (not a hard choice with Uber).
That's a common misconception. If I loose a leg, the driver will probably never be able to pay the sum, go through personal bankruptcy and leave me with a nice sum I sued for, but no actual money.
Suing takes time, and even if you succeed the person/company might not be able to pay. Insurance companies pay out immediately if there are medical bills to pay.
"Insurance companies pay out immediately if there are medical bills to pay."
That's not true at all. And it REALLY depends on which state (in the USA) you are in. Florida for example is a no-fault state. Which means, basically, insurance companies are on the hook for the first 10k. Which is about how much it coast to mend a broken leg. Anything above that requires a lawsuit. And you don't sue the insurance company. You are suing a person (or company).
Lawsuits take time. Lots of time.
A taxi driver in Florida has different requirements. So does the company he drives for. The amount they pay out is much higher. Also, not every issue requires a lawsuit. Sometime you just get a bad driver. With Uber, who you gonna complain too? Uber?
In most states, complaining to the taxi commission is a serious offense and has to be answered by the driver and company he drives for.
I don't have a problem with Uber itself. If people want to drive others around, fine. BUT totally unfair to require taxi drivers to be licensed and not the Uber drives. It's unfair to make the taxi driver (and company) have much higher insurance requirements than Uber and it's divers. And it's unfair to the rider. If you are in an Uber car and you have a wreck; you better hope that driver has more then "what's minimum" for insurance. Chances are, he doesn't. And you're fucked.
PS: This is true for car services, and even courier services too, by the way.
No, public healthcare makes sure you get on your foot again an then tries to get their money back as any other insurance would.
So, the public makes sure the damage of you being hurt, unable to work and maybe turning out to be a case for social security because of being maltreated is avoided.
You or lawyer takes the cut after winning. Case like that will make Uber to require insurance from all drivers. Problem solved without extra regulations and bureaucracy that will cause problem in the future, as we have now.