I don't really find this to be the case at all. I've had Waymo for ~2 years now (since the private program), and I've never noticed it being quicker or cheaper than an Uber. I have several hundred rides; I prefer the service - but I've never once told people it's cheaper or faster.
Currently, on Wednesday March 26th at 8:34 a ride from Bar Part Time in the Mission to Verjus in North Beach is $21.17 with a estimated 8 minute pickup time. The same ride on UberX has an estimated 2 minute pickup time at a cost of $15.34. I could see it being cheaper if you top 20% - but I don't tip nearly that high on Uber rides.
I will admit that I could possibly be self-selecting to peak times as I own a car in the city, so I only use ride share in the evenings; so it may very well be the case that the price/wait is more competitive at off-peak hours.
Furthermore, it's quite surprising to me that it seems that the human labor cost doesn't affect the price at all. The only price controls seems to be demand and the latent demand is enough to create a price floor where there is always a human that is willing to drive. It also seems like plain old logistics and traffic will prevent Waymo from providing enough supply to offer dirt cheap rides. The fact that a ride that would have cost me $5 in 2016 is almost 4x as much with "magic self driving technology" is not something I could have told my 2016 self.