Uber is the ultimate middleman company. It would only work if people bought their own self driving cars and rented them out to Uber to make some extra money.
At which point of course owners would shop for top dollar and riders would shop for rock bottom and they'd still be stuck lighting money on fire to subsidize rides.
And it's difficult to see how those ad hoc rentals can compete with a fleet management company that has economies of scale and optimizes for short-term rental. Car depreciation is more mileage-based than time-based so it's reasonable to assume that the rates people would get would tend to reach a point where they were pretty close to the cost--with perhaps an exception for times when there's extraordinary demand.
Yes and no. I can certainly agree on a theoretical basis, but the black cabs that have crawled around London since the 50's are privately owned and built to last.