It makes no sense to own and maintain your own FLEET of self-driving cars for your own family (since it's virtually guaranteed that multiple family members will need a car at the same time, at some point in time).
In the suburban sprawl where the majority of the USA lives, we already have a nicely distributed network of storage (homes with attached garages). That would be far more efficient than having Uber and Lyft build their own duplicate storage for cars, which would almost certainly put them further away from the customers who need them and be far less efficient.
I know this is HN and we're all supposed to be just super ga-ga over the idea of one single company owning the entire future of transportation, but there are many more possible scenarios than just "everyone gives up control and ownership of how they get around".