I've speculated on this before...
I expect that subscribed-to sedan services will increase in popularity as a step above taxis. Paying more than a conventional taxi and giving the subscriber the ability to report/reject cars that are in poor condition will allows the service to charge and ban offenders that mess up cars. On top of that, there are services for school buses where an on-vehicle camera system records the trip to a local disk only and overwrites the recordings after so many days unless a report is made that the footage needs to be pulled before it's overwritten, at which time it's retrieved over-the-air when the vehicle comes in for regular service at the company's garage. That system would work relatively well for a subscription car if it doesn't catch audio and isn't pulled unless there's an actual reason to pull it (like vandalism or evidence that the interior was used for a crime) so long as such conditions are made clear from the beginning.
Taxis will still be a thing, for either those that don't need a sedan often enough to justify paying for a subscription, or for those who cannot subscribe to a sedan service due to previous behavior. Used like a service they'd probably cost more, but used infrequently it wouldn't be that big of a deal. There would also be a greater likelihood of recordings being reviews more frequently.
Private car ownership will continue for people like me that have plenty of room for parking and like you, don't want to share the vehicle with unknown others. I look forward to scenic road trips where I can look at the scenery instead of always having to drive, though I would probably want the option to drive. It would be convenient when going to congested places to be able to be dropped off and let the car go find a place to park itself, or even for the car to just go home if the per-mile cost (like for an electric) is low.