I don't think Uber in its present form will shift the demand curve dramatically. Driverless cars will. A world with driverless cars could follow two models. Let's call them Apple and IBM.
Apple: each household owns a driverless car. Perhaps it can be occasionally rideshared via UberX, Lyft, etc. In this world, you are right: Uber tomorrow will be Uber today.
IBM: Uber (or Google or Tesla...) owns (and finances and repairs and replenishes) a fleet of driverless cars. Households pay it use fees. In this world, Uber's strategic autonomy is dramatically improved over today. If I were Uber, this is the long game I'd be playing for. I would be (a) training riders to depend on hiring cars with minimal personal contact, (b) encouraging the development of driverless cars, and (c) building a cash pile for rapidly acquiring a vast fleet of driverless cars the moment they become viable.