In a world of ubiquitous cheap taxis, you don’t need nearly as many cars. Since not everyone commutes at the exact same time, any given car can shuttle several different people to work. And the average commuting vehicle can be small enough for a single passenger to sit in comfortably. This smaller number of smaller cars could get by with radically fewer parking spaces.
I just don't really see how you get around the peak commute problem. Sure not everyone commutes at exactly the same time, but despite/as evidenced by peak time congestion people do commute at the same time to a surprising degree. So, yes, sure, the car fleet could be reduced some and parking lots could be at the very least placed elsewhere. But there are good reasons people don't like carpooling which wouldn't really be solved with this type of system.
As long as there's a peak commute time, we're gonna have a lot of cars.