In the future, let's imagine in 20 years, we may see "cars as a service", so essentially you can summon a Tesla, and the Tesla can bring you anywhere.
In that perspective, isn't it clear that the liability is to the company that organised the trip ? In similar way to when you get carried by a bus, a taxi or a train ?
One problem I‘ve always seen with the fleet model of widespread automated cars is that in 2020 cars are very much status symbols as much as they are modes of transportation. People own BMWs and Porsches to show off status first and get around second, and these types of people will always want “their” car.
Of course there are still plenty of edge cases where individual car ownership is nessesary, whether that's because too few cars of that type would be booked to make a fleet viable (Rolls-Royce) or because you are in a very rural area.
Rich people who want to call a chauffeured luxury car can do so in most major cities. Many of them do use such a service, but many of them also own expensive cars to show off their wealth.