Earlier quoted context omitted.
If a couple percent of cars were in this kind of system, they'd be waiting all over. It would be much less than 20 minutes. They could even set up equivalent "stations" where there's almost always a car parked within a couple blocks and it can get there before you do. I don't go on a lot of long-term trips, but if I did that would still be a reason to get one car for a household instead of multiple. For rarer trips I…
Really it sounds like you describe an efficient public transport system :). Of course not every city has that (I hear public transport kind of sucks in the US), but still it's important to know that other countries have that, without autonomous cars.
But also those busses wouldn't be able to come to your door, or drop you off directly at your destination.
So I don't think I'm describing efficient public transport. I'm describing something like the platonic ideal of taxis in a dense city, but without the huge number of drivers.