Uber's competitive advantage now is their mindshare, app experience, and subsidized pricing. With or without self driving doesn't really change that.
If you have too much demand and not enough supply, then wait times shoot up, and people return to your competitors.
Plus demand is non-uniformly distributed across time. If you have enough cars to handle peak demand, then they’ll be sitting idle the remainder of the day/week.
Most people think that trains are busy, because most people ride trains in high-demand times of year. If the same is true of a TNC, then it’s much better to list your cars with a bigger network and let them keep your cars busy more than you can.
It’s much more expensive to properly balance the marketplace than you think.
I think the last one is their big problem: Uber has their current market-position by selling below cost, and they've locked in a lot of things like compensation at the assumption they were profitable to a much more fanciful degree. Once the VC money runs out, they're going to need to raise prices and it's a lot easier for anyone else to compete with them since they're all renting the same Waymo cars and all of the drivers are running multiple apps anyway. Uber has some economy of scale benefits but they also have a reputation for mistreating drivers and a lot of expensive real-estate and staff which local competitors don't have.