Ideally yes, but there's a lot of hurdles to overcome. For many it would work just fine, but for other there's going to be minor details that makes it work less well.
Firstly we need to spread out peoples work day. We can't all be at the office a 8:00, and go home at 16:00. it wouldn't make economical sense in have a fleet large enough to handle peak hours, but have the cars do nothing most of the day.
There's also details like: I would like a car to pick me up at 15:45, and drive me to the kindergarten, and that car MUST have a seat for a three year old (but not a seat of brand X, Y, or Z, because they're to small). Or I need a car with a trailer, or a truck, because I'm moving 500kg of dirt to the recycling station.
Most people shop around for cars and pick very specific cars that fit their needs. So renting for a fleet of cars also need a rather large variety and flexibility in allowed usage.
I believe that many would still want to buy their own cars, even if it's self-driving, to ensure it's available when they need it, and that it fulfil their specific needs.
If we didn't have all these specific needs, most of us could just take the bus, which in turn would make it feasible to have all the bus routes people would need.