I'd tend agree with your comment in the short term. Moving my daughter's child safety car seat would be a pain, for sure. Longer-term I would expect to see modular standards emerge for child safety car seats. You lock your child's seat into the standardized base that's hiding under a seat cushion in the back of the car. There are already stroller / pram systems that incorporate the child safety car seat, so I could expect that market to boom, too.
Moving things besides yourself is something a lot of people in urban areas already do with public transit. With a private autocar service it's more likely that you'd get more space than on a subway or bus, so I'd think it would be even easier to move more than just yourself. I would expect a rental option to let you keep the car with you, as opposed to it returning to the fleet when you reach your destination.
The model I see that will remove individually-owned, occupant driven vehicles will some variation of:
My company will purchase your auto and credit you some amount of haulage on my autocar network, and perhaps some cash. For the low income people using sub-$1,000 cars a token amount of credit will be sufficient. The real value for them will be the convenience and reliability of transport they get from my system. Imagine not losing your hard-won job because your "beater" car failed at an inopportune time. Imagine not having unexpected car expenses that make you choose between, say, food and transportation.
The higher-end market will be a tougher sell, but there are likely productivity, personal time, and environmental impact arguments that can be made there.
I'll sell your car (likely at a loss) in some area when I haven't achieved market penetration yet (or for salvage). Once you're bought-in to my network you might leave me for a competitor, but ideally I'll use regulation, contractual lock-in, and other tools to attempt to keep Customers tied to my product until I've recouped my initial investment.
I think it could work. I don't particularly like it, but I think it has at least a non-zero possibility of happening.