Costs may well have shrunk to such a low level that it isn't a big deal to produce a number of vehicles to meet peak demand. Fully automated driving doesn't necessarily mean a ban on personal ownership but shared taxis would be the natural choice for most if they are available at much lower cost, with negligible waiting times and no worries about parking.
No matter how cheap these Robotaxis become, no company is ever going to have enough to cover periods of extremely high demand; you'd simply end up with thousands of cars twiddling virtual thumbs and not making any money for their owners (as well as creating congestion). There will be enough to cover routine maintenance (if they don't skimp on that), charging and perhaps a handful of surplus vehicles to give a capacity boost at known peak periods, but don't expect there to be a massive increase in the number of available cars when there are large events (football matches, concerts etc.) taking place. Those companies will simply whack up their prices at peak times.
I'm not sure that you'll have negligible waiting times, either. In an ideal world you'd have a car patiently waiting at the end of every street; in reality, those cars will be heading back to locations where the jobs actually are, as happens with ordinary taxis at the moment.
Parking bans may well be the thing that kills personal ownership, as it will seem incredibly wasteful to sacrifice road space and land for parking.
It will be a long time before it kills personal ownership; it may become viable for people who either live in or close to city centres but I can't see it being an option for those in suburban or rural areas.