Uber just needs to think of themselves as a mobile staffing company and they'll be fine. Unfortunately, staffing is a market where "Move fast and be an asshole" isn't a great philosophy so they'll need a bit of a temperament change to survive.
This would basically turn them into a private driver service and finding qualified staff would be a pain for anyone wanting a french tutor for the drive. Ignoring that as an off the wall example though they'll still hit a pretty hard wall with trust. Even if the fear is unfounded trying to get a huge number of people to sign up for random (to them) people to pick up and drive around their kids is running into a lot of child abduction/abuse fears that run kind of deep in the US.
Why? Just filter on the "Speaks French" tag.
Are you trying to argue that there is zero use for a person with a car who speaks French in a city like San Francisco? I really don't understand what you're trying to convince me.
Why? Because self-driving cars are basically a fleet service driven by a software. Once you remove the driver (where Uber spent so much acquiring) The only differentiator is the consumer facing experience. Neither Uber or Lyft will have as much power as Apple and Google since they ultimately own the mobile experience.
I ultimately envision this business as kind of a Kayak mobile on the phone managed by Siri or Google/Apple maps that will call the nearest taxi or the cheaper rate aggregating from multiple possible vendors. Larger fleets (Uber, Apple, Google) to smaller individually manage fleets. Car companies might decide to also enter that market in collaboration with financial underwriter.
So having a network of drivers ultimately gives Uber some leverage and removing them from the equation, I think it'll actually destroy Uber. This is why I think, this could be a blessing in disguise.