Your estimate of 20 cents per mile is rather low I think. Even though there is no driver, the owner of the car still want's return on his investment, but sure for a lot of people it might be a perfect fit. And like you say they can scale the cars and deploy according to demand, which can save on energy consumption. On the other hand if we had to use such a service to do our shopping, the car would have to come to our remote location first, then drive us to the shops and back to our home and then on wards to where ever, it consumes more energy then us having our own car and drive back and forth once a week. (Yes we try to minimize our shopping to only once a week. Good for our wallet and the environment.)
This is why the scale of the service is really important. For a commute, a one-person microvehicle could easily use only 5~8kWh/100km and maybe fit just a 15-20kWh battery on board. (It would, of course, be electric. I don't see any reason these should be powered by fuel.) With downtime every few customers for recharging or battery swaps, that vehicle can be utilised something like 16 hours a day, handling commuters in the morning, business travellers in the afternoon and return trips towards the evening. Even if it costs $50k to buy, if it lasts 10 years and does an average of 20mi/h in its service time, then it only cost 5c per mile, so the rest is just down to how much energy and maintenance it requires.
For your shopping example I see it going one of two ways. You would probably only need a one/two person vehicle to go to the shop, and a larger vehicle to go back. A supermarket could convert most of its parking lot into a dropoff/pickup area with vehicles waiting to return customers home. If the service is sized right then there's always a use for that return trip. Alternatively, the delivery vehicle could just take your goods home a few hours after you paid for them. That's much more efficient because the chances are, the route can be arranged to maximise efficiency with other customers served as well. Since a considerable amount of space would be freed not having to accommodate every customer's parking need for the hour they are there, the store could expand to fill the space, or other businesses could be co-located on site.
The whole idea is that you don't need the vehicle to be the same size all the time - it would adapt to your specific needs.
In this hypothetical world I still think private cars will be a 'thing', but they will be less common. Whether you will be allowed to drive as a human though is another matter which I'm sure will create plenty of debate.