What does Waymo get out of this? It seems to me like if the value proposition in self driving is there (privacy, consistency, etc), people would be willing to download another app? Especially once word of mouth goes through.
Plus currently the Waymo app has neat features (when the car is on its way to you, it shows you stuff like when it's stopped at a red light or whatever) that Uber probably can't replicate due to API access issues.