No, it actually won't.
There is nothing that a driver would do that an automated taxi would not, because both the driver and the automated taxi are optimizing for their ability to give as many rides as possible for as much money as possible.
The main reason to think their behavior would differ is on the margins where drivers have to go home and sleep and robotaxis do not. But that really makes it likely that a robotaxi service will keep its cars out 24/7 except for maintenance.
The other reason to believe drivers and robotaxis will behave differently is if you think drivers are leaving money on the table, or robotaxis will. And Silicon Valley simply does not leave money on the table. I'd be more ready to believe the drivers do.
There's another other factor here: when you make cars more available and more convenient, people are more likely to take them. So robotaxis will probably increase, not decrease, road usage, if only because they never need to sleep, eat, or go to the bathroom. They will just repeatedly get a fare, drive the fare, and go do whatever they do to be idle. Which, in many cities, is going to be "wander the streets" or "take up parking spots in the city center".
EDIT: I had responded to this without reading the OP article. The article has some good discussion of the kinds of secondary effects that are likely to cause robotaxis to increase road consumption.