"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.