Many, many people already do this on public transport. What we're talking about is general door-to-door public transport.
Car-pooling is awkward to arrange, particularly for short joined parts of the journey (e.g. picking someone up part way and dropping them off a little later), and it's extremely hard to arrange anything that overlaps such that the first passenger needs to get out before the last passenger (currently would involve handing over the keys to your car).
Total fantasy. Virtually nobody uses public transport. 5% of commuters use public transport. Relative to the car-pooling, who sees double the volume of users as public transport, 10%.
The question is: if that changes, what might people choose to do? The example of large cities like NYC at least teaches us that the American preference for solo car travel is not fixed.
[0] http://ns.umich.edu/new/releases/21923-hitchin-a-ride-fewer-...
http://www.ons.gov.uk/ons/rel/census/2011-census-analysis/di...
http://www.ons.gov.uk/ons/rel/census/2011-census-analysis/di...
These are not insignificant figures.
If you say "people don't want to do X" when millions of people in my country already do a more extreme form of X, then I think you're wrong.
This will be much smoother if you just open an app like Uber and it tells you the next pickup time for where you want to head.
Those people might use a vehicle subscription, because the cost of private rides would still be lower than the costs of private vehicle ownership (depreciation, maintenance, insurance). Many people would pay for the privilege of a private ride (like they do with Uber/Lyft now).
If car subscriptions services succeed and are nearly ubiquitous, the personal-freedom symbolism of car ownership will degrade.
Then, if people want to own a car as a status symbol, they will probably be better served by a more expensive one anyway, the same way people buy other status symbols (rose-gold iPhones, etc). Ironically, most of those status-symbol cars will still have an autonomous mode.
Some people pay more to use a first-class compartment.
Shared-car design could also be drastically different. I don't want to share a Honda Accord with random people, but I wouldn't mind sharing a road-train that had private compartments for each set of passengers...