The problem with your logic is the connection to Uber. Uber doesn't control the cars their drivers use. Sure they can set restrictions but they never collect actual data that is useful for self-driving. The only data Uber collects is when and where people want to travel. Google & Apple both have this dataset (probably superior) through their map apps.
The real data gathering will come in the way Tesla does. Gather data from real-time drivers by putting the necessary sensors in cars you sell.
Maybe Uber could force their drivers to install kits but I doubt it. Self-driving cars will emerge over time as new features in cars you buy. This is already happening (ie brake assist). You'll then just get updates to your car when new features become available.
There's a pretty good talk about the future self-driving cars here: https://www.youtube.com/watch?v=iwcYp-XT7UI