Keep in mind your taxi service is cheap because it's on the road all the time. A taxi might make $250/day, and pay the driver half of that cost. That means the company is making $125/day per taxi. You're one of 15 or 20 people taking that taxi in a day, so the $15 trip is possible. If everyone was taking an automated car to work, they need to pay the majority of that $125/day price. Split it with 4 people, and that's just over $30/day each. Kind of expensive. Add more people and we're turning into a bus or public transit again.
Work schedules offset would help everything all around. If some people worked 7-3, 8-4, 9-5, 10-6, 11-7, etc, then we could decrease rush hour traffic, and use the cars multiple times to lower the price to $10/day, which starts to become interesting.
I like your idea of buying a car, adding it to a network, and getting money back when you're not using it.