The car sharing won't cover everyone's needs and some people will stick to keeping their own car. I'd imagine people living in urban areas are more likely to use a car sharing service since it's uneconomical to own a car in most cities. I'd imagine the suburbs will be a mix of sharing and ownership. I can imagine a family having a car they keep for weekends or whenever the parents and kids need to go somewhere but they still use a car sharing service during the weekdays when mom takes the family car to work and dad need to get to work too.
I think you're exactly right. Once cars can drive themselves to destinations, it becomes so much easier to get by with one dedicated family car, supplemented with car sharing services.
For example:
Mom takes service to work, possibly carpooling. Dad takes uses family car to drop off kids and goes to work. Car drives itself to mom's work for her to use to pick up kids and commute home. Dad takes service home. Or whatever. The point is kids can always be in the car with the extra diapers, with safety features parents want, etc.
The other part of this that people ignore is that a car sharing service could open up a lot of flexibility. For example, maybe our dedicated family car is just a sedan, but on this particular weekend with the grandparents in town, we can use the car sharing service to get a minivan and have everyone ride together. Or I want to pick up a piece of furniture and request a truck from the service. Or it's my anniversary, and I can request a luxury sedan to drive us to the restaurant for a touch of opulence. I'm sure there are many other examples people can think of.
Like you say, car sharing won't cover everyone's need, especially at first, but eventually, the big players in the space will have enough sophistication to cover many cases, especially for a second car.
I think car sharing even works in those cases. It's all a logistical problem.
Not that there won't be some people who own cars. There just aren't that many problems that are hard to solve with car shares to outweigh the huge benefits.
car sharing works less well for getting to work because everyone else is trying to get to work then. You may as well own your own car for getting to work because you pay for all the costs anyway. All car sharing buys you is fixed maintenance costs on your car.
Car sharing starts to make sense when you want the car outside of rush hour. Want to go to the store for an hour at 2:30, no problem there are plenty of cars free. Want to get to work every day - you are paying peak rates for a car that will probably be used exactly twice that day, once to get you to work and once to get you home.