I completely agree. For-hire self-driving vehicles are the future, I just don't believe Uber and Lyft are going to be the platform.
Off-topic: Whenever I see analysts argue that Tesla isn't making cars fast enough to compete with Big automakers, I chuckle. Elon doesn't have to replace every car, he just has to replace the door to door trips, which he can do with only ~10-15% of the current US vehicle fleet on the road. Self-driving Tesla vehicles for hire would further fund Tesla expansions (which provides additional capital for their Gigafactory to drive down battery storage costs, thereby enabling the wide-spread transition to intermittent renewables such as solar, wind, and tidal).
One reason to take a cab, or uber, or a train to the airport or downtown is the time and expense of parking. But if my car can just park itself at a low cost remote structure, go home, or say earn a few bucks taking a fare, and come back for me, then it's less of a hassle to drive to the city. I don't have to worry as much about being too tired to drive longer distances, I could just snooze or read a book. I'd probably be more likely to let my car drive me to LA overnight, and perhaps take fewer flights and consequently rent fewer cars. I could send my car to pick up food or packages or friends. Overall, it seems like my car is more useful to me, not less, if it can drive itself. If I can hire it out when I'm not using it, it might make it more affordable to own, as well. I doubt most people would rent out their own car with any regularity, unless they had to, any more than they want a bunch of strangers sitting on their furniture and messing with their stuff.
While personal purchase of self-driving cars will be a thing, one big value proposition of self-driving cars is a drives down the relative cost of cars-for-hire compared to personally-owned-vehicles because you no longer need an operator in the vehicle whenever it is available for use (which means that the per-trip fee no longer needs to be enough to pay not only for the maintenance on the car and the operator's time on the trip, but also the operator's down time "on call" to be ready for the next trip.)
Uber might not win the space, but I think its naive to think that self-driving cars don't shift the balance more in favor of for-hire cars.