I've never really understood this.
Their drivers do more than just drive, don't they? They also supply the cars, and deal with maintenance of those cars. If Uber switches to a self-driving fleet and ditches human drivers, who will own the cars?
If Uber owns or leases the cars, Uber will have to be dealing with maintenance. I suppose they could go to a model where people who have bought self-driving cars can let Uber use them when the owner does not need them, but then Uber has just replaced paying a zillion drivers with paying a zillion self-driving car owners.
So, assuming they go with the model where Uber is the one handling maintenance so they don't have to have a zillion pseudo-employees I don't see what would stop other companies from doing this better.
If Waymo wanted to go for that market, for instance, they have better self-driving technology, and could probably get integrated with the Google Maps app that is already on everyone's phone so they wouldn't have to convince people to download anything. Sure, Waymo doesn't currently have, as far as I know, anything set up for managing and maintaining a large fleet of cars...but neither does Uber.
Or what about the car makers? Many of them are working on self-driving, too. I could see some of them deploying self-driving taxi fleets, with maintenance handled through their dealers. This could become a new sales channel for them--new car to their taxi fleet for a couple years, then sold off as a used car which the buyer can be assured was well maintained and still has a good warranty.
Imagine your self-driving Audi/BMW/Mercedes/etc taxi dropping you off at home at the end of the day, and then presenting you with a report showing how much you used the service over the last year, how much it cost you, and how it would have been cheaper for you to have your own private self-driving car, and offering to sell you that very car. If you accept, the car drives back to the dealer, gets cleaned, inspected, tuned up, and drives back to your place to be waiting for you to take ownership the next morning.
In summary, owning and operating a fleet of self-driving taxis is sufficiently different from what Uber does now that I'm not sure that Uber has any particular advantage over anyone else who might want to do so, and might even be at a disadvantage, so I'm skeptical that this is their future.