Here's the parts of your post that make no sense.
"city (or licensed companies) will maintain"
Why on earth would individuals not be allowed to also own personal self-driving cars? Why can a car not have two modes of operation, self-driving in self-driving areas and manual in other areas, if the user so wants? Why would everyone be just fine with not being allowed to have a personal vehicle that they can leave their stuff in between rides, meets their personal style preferences and transportation requirements, and wasn't beat up or graffitied or left smelling like a dumpster from the last unknown occupant? What's the logic in *making* everyone use shared vehicles if many individuals logically still want to own them and they meet all of the transportation requirements? None, that's what. It's a pointless exclusion and one which would render your system unacceptable to a large portion of the population.
you'd never use it living minutes from downtown anyway.
Right, so a guy with a degenerative muscle disease or a grandma who can barely walk to the mailbox are just going to walk? Sorry, but this "it works for me so it must work for everyone" attitude is ridiculous. And even for people who are in perfect shape it's not always a good option. I live in Reykjavík where we sometimes get surprise blizzards and hurricane-force winds happen usually a couple times a year. Am I supposed to walk to work every day?
Or if you mean "everyone's supposed to use these shared city-only cars": again, maybe that suits your life, but here in the real world, a large chunk of the population isn't so city-bound. I head out to the countryside about 5 times a week or so. Am I supposed to take a taxi every time?
Yes, an effective automated-driving system opens up great new beneficial opportunities for ride sharing and would be used by many people. But your "all cars in town must be city-owned public transportation" concept is ridiculous and would never be accepted by the general population. There's not even a point to your proposal, as privately-owned self-driving cars fit just as well into your scenario, it's simply a capricious exclusion on your part.
People use vehicles for all sorts of things. A number of times in the past several weeks I've used my car and my truck as a flashlight. I don't have good outdoor lighting at home and on my land in the countryside I don't have power yet, and daylight is in short supply this time of year, and so to work outside, it's the most logical solution.
My car is a 2-seat first-gen Honda Insight. Why? Because I don't like today's higher-drag trends and it's more than is needed for my daily commuting needs. A lot of people however would find my choice hideous and not want to be seen in it, and it doesn't even begin to met many needs that exist. I would be uncomfortable having to take a high-drag car to work every day however in order to meet these peoples' preferences and needs. There are so many thousands of types of cars in order to meet the general populations' many preferences and usage needs. That's not going to suddenly change. In many cases, people are indifferent. Good, automated rides from shared cars are perfect for them! In many cases they're anything but indifferent.
I use my truck to haul supplies around my without-roads land. And let me tell you, unless the algorithm is smarter than me, it *cannot* do that better, because it taxes experience to know where the ground is unstable or so marshy that it'd sink in. A mistake could require a thousand-USD crane rental or worse leave me crushed dead at the bottom of my canyon. I wouldn't even trust it on my driveway on my land, as it hasn't been fully built out yet and the bottom can scrape the undercarriage in places and is unstable in others. These sort of things aren't just simple decision making based on distance measures ultrasound sensors and transponders. They take image recognition, understanding, and complex reasoning.
My insight's engine is dying. I'm fine and used to it, but many people would be really ticked off if it just randomly showed up to give them a ride and struggled to get up a hill.
My pickup is loaded with stuff in its backseat. A lot of people would say, loaded with junk, and be really annoyed if it just showed up. But when I need various tools or parts, I have them. I can't simply do that with a shared vehicle.
I could go on, but you get the picture. There are perfectly rational reasons to still want to own private cars, even in a world where cars can drive themselves. Such a world makes it easier for people who don't want to to not have to own a car, but it absolutely doesn't mean people shouldn't be allowed to.