A rail line can move as much as 20 car lanes today. Even if you double the space efficiency of cars with automation (a long shot considering you still need to accommodate for pedestrians, cycles...), transit will keep being vastly more efficient.
Automated cars will optimize parking and make on demand transit more affordable in the suburbs, but it will not solve transportation in dense cities.
LA is a dense place on average. The greater LA is more dense than the greater NY.
Currently we have about the worst utilization of vehicles possible. Drivers drive alone, and park their cars for long intervals. This is why one can claim that trains move enough people to fill 20 car lanes. At the other end of the spectrum, diesel busses are noisy, smokey, need a professional driver, and go from underutilized to packed, depending on time of day.
A mix of automated vehicles will be able to much more closely approach optimal load factors at all times of day. 6 or 8 passengers can ride in a car-sized automated vehicle. That makes streets a lo-capex alternative to rail.
Automated or not, mass transit is just geometrically way more efficient, there's no way around it in dense urban environments.
Your argument might be valid for a mid-sized american city, or a suburb, but it doesn't work for a european city, or even a dense american downtown.
Actually diesel busses are only "efficient" when they are full. They're barely as efficient as single occupant cars just on fuel use when you factor in the bus running underutilized much of the day. That's not counting the cost of the driver, and assuming bus capex is efficient.
So even a commute-sharing system like that being tested in Waze would be more efficient than busses.
Buses take as much space as 3 cars. Even if you can manage to optimize everything perfectly to pack 4 people in these cars at all time, you only move 12 people using the same amount of road of a bus transporting 60 to 90 people, much more in articulated vehicles. There is just not enough room in dense cities.
You have to consider the overall efficiency of a bus network, not one particular trip. In urban centers where buses are run on frequency, not timetable, a transit line is attractive because it runs often. People take the bus in rush hours in part because they know they can depend on it all day. These seemingly low-efficiency trips where the bus is half empty are actually very important to the overall efficiency of the network, because they induce demand.
The case for transit is a space argument, not a fuel argument. Mass transit is simply the most space efficient way of moving people around.
Uber/Lyft are still very much premium products. People earning minimum wage cannot afford to Uber to and from work every day, so they will be relying on public transport for a long time yet.
Uber certainly helps deal with the lack of public transportation but the other big issue is LA is HUGE! My sister lives in Glendora (far east side of LA) I was staying in Venice Beach. It's about 45 miles. And that's not even one side of the LA metro area to the other, that's just LA itself. On Uber that would be ~$55 or so. A similar distance on the train system in Tokyo would be $10-$20 depending on how many different companies' trains you have to use.
Of course it might still be cheaper than a owning a car if you're not making the long trips often.