I think it is a big assumption anyone will be able to buy a bunch of self-driving vehicles and blanket a city. Tesla has already prohibited auto-pilot (or whatever it ends up being called) from being used for third party ride sharing services. I would suspect other manufacturers will too barring exclusive contracts with the transport/ride-sharing/etc company.
This means right now it is an arms race between Tesla and Uber to manufacture, at scale, the first fully self-driving "taxi." Tesla is probably well ahead on that game and Uber will either need to partner or buy an existing car company.
The secondary problem is the platform is the mobile phone. Apple or Google could easily build an auction platform in to the OS for calling vehicles. In that case, it could become a commodity business with royalties paid to whoever has the dominate mobile device in the future (could be Facebook or Snapchat instead.)
Thirdly there may be vehicle differentiation. Right now if you use UberX, there is a pretty wide range of vehicles that could show up. If brand preference and car features matter a lot, I don't see why a luxury car maker like BMW couldn't at least a niche player.
Finally, everyone has different ideas. What this looks like in 12 months is going to be different that 36 month, vs 10 years from now. I think some people are right about their assumptions of what will happen, but maybe it will only be that way for a few years before completely changing again. How it plays out probably has more to do with what decisions and deals the big players make -- Uber, Tesla, Google, Apple, the other car companies -- more than any particular inevitability.