If I can send my car out without having to drive it myself, I'll use it for things things like picking up kids. Take my spouse to work, drive back home, take me to work at a different time, repeat.
Right now many families buy two cars, often not because they need to drive both of them at exactly the same time, but because they need to use them in different places. If you can send the car between locations then you can use one car, and the cost is that the car spends time on the road, making someone else's commute worse.
Again, this is Uber's entire business model, and if they're successful at getting everyone to use ride-sharing, regardless of self-driving cars, then we will trade a bunch of parking spaces for more cars on the road. It's literally how Uber works.
In addition, if your car can drive itself places, whole new business models will spring up around things like delivery of food, packages, etc. Businesses will be happy to start providing you the service of putting stuff in your trunk.
Yes, I'm guessing that the arrival of self driving cars coincides with electric drive becoming commonplace. They could even go hand in hand: send the car back home to charge, then have it pick us up later.
All of these things reduce the incremental cost of having your car on the road, which will drastically increase traffic.
Most errands involving cars also require the user on both ends to load and/or unload or perform some other function. But, sure, we can ignore that for the sake of argument...
> or if hired cars can drive halfway across the city to pick up the next fare with no impact to the car's owner besides wear and fuel.
Well, it will if there is no cost other than low-cost fuel (like the US has no) or similarly low-cost energy (e.g., for electrics).
OTOH, the solution to "the aggregate amount of traffic is too high because the cost is too low" is to increase the cost -- probably not with gas charges because of the move to EVs, but by tracking cars on the road and charging for their use of the road, more at peak hours.