I think the point is price.
Cabs are expensive, but most of the expense is paying the driver. Once you get rid of the worker, it gets a lot cheaper.
Also, with centralized control, routes can be optimized so the taxis are always driving and carying passengers.
It's not slightly cheaper than driving, it can potentially cost an order of magnitude less, and be faster.
Where I live, public transport is easily 5-6 times less expensive than driving, combining bus and cab rides, including the labor cost of drivers.
About keeping it clean, and accountability, we are now used to be identified always. The cars can have cameras, and even require an id for you to ride them (they won't be taking cash, after all). There would be abuse, but it would be close to trivial to punish that kind of thing.
If there's vomit in the car, the car should be able to detect that, and go to a cleaning station. In the event that you do get an unsuitable car, you can just reject it. You could even look at a stream of the inside of the car before it gets to your home.
Also, think about the carpooling possibilities. While people don't like sharing space with strangers, price can change some minds.