Does it matter if the driver was in a Uber, Taxi or a personal car? It was a regular human in a regular car.
Even though a taxi driver is pretty much a regular human, and a crown vic in taxi-yellow is pretty much a regular car.
Example: High-risk oil drilling operation off the coast. Engineer drops a wrench, wrench breaks a pipe, thousands of gallons of oil flood into the ocean. Do we hold the engineer liable for a billion dollars in environmental damages, because he just dumped oil in the ocean? Or do we hold the company liable, because they were engaging in oil drilling that inherently risks dumping oil in the ocean?
Does it matter if the miner was an employee of International Coal Group, Inc, or if he was digging in his own backyard? It was just a regular human in a regular mine explosion.
The point of these counter examples is that accidents resulting from industrial activity are certainly morally tied to the corporation that is profiting off said activity. Uber is taking regular drivers, and putting them in a totally non-regular working environment. There are psychological effects to sitting in a drivers seat for hours and hours that regular commuters don't experience, yet Uber is not training their drivers for this.
Being responsible for the accidents your economic activity encourages is part of the bargain of Incorporation. To operate otherwise would encourage trading safety for profitability in the most extreme ways.