What's going to happen and is already happening is cars will offer certain self-driving features. Things like lane-keeping, brake assist, highway driving. Car companies will also put in all the necessary sensors to collect training data. Then provide over the air updates. Tesla & GM(Cruise) are already doing this.
There's not really a way for Uber to get to level-5 first and at that point, why build a kit that integrates with your old Honda to give it lane-keeping?
The only dataset Uber has is a general sense of where and when people want to travel. Google & Apple probably have much better data through their map apps. They both also have an ungodly amount of money to both acquire the best researchers and create & distribute these kits.
In my opinion, self-driving cars are a race between auto companies and these cash-flush tech giants. Uber/Lyft/Didi all don't really have a chance.
The auto companies and their suppliers like Bosch. And, if Apple, for example, were to come up with a really good system for autonomous driving or a subset thereof, don't you think they would be happy to sell it to automakers who already buy a huge number of component systems. After all, they already do with CarPlay. (Which is better than any automaker system I've used.)