That's a possibility. I'm not sure it's true. Both Uber and Lyft have good waiting times right now. The hypothesis that they will replace privately owned cars supposes that the market will grow immensely.
Waiting time is a function of: number of cars, size of the city, number of customers (to ignore congestion for now...) To have a good start as a new entrant, it's just a property of the size of the city. Then, you need to grow your fleet with your number of customers to keep waiting time constant. That number of cars to start with, the barrier to entry, is a tiny portion of what you need to add to keep up with demand.
If you don't have to compete with drivers, with autonomous cars, it's not that hard to meet and start a price war.
Again, I don't see how Uber will avoid being squeezed out of their margins the way airlines are. It's not cheap to start an airline, yet many people have done it. And all you need is 3 competitors to drive prices, and margins, down.
