Most people who use mass transit use it because it is the most efficient way to get from A to B, not because they can't afford their own vehicle, nor because it's the cheapest option.
Case in point: I stayed in Atlanta for a 4 day weekend at a convention downtown. I drove to my hotel, then used the hotel's free airport shuttle to the airport to take the subway/train system MARTA to downtown Atlanta and back daily (sometimes 3 or 4 round-trips in a day). It cost me all of $10... and it was the fastest way to get from my cheap hotel to downtown as there was also a ballgame and another convention as well and the roads were bumper to bumper. I rode the train several times a day - got my money's worth and met interesting convention-goers on the train. I took a taxi back to the hotel one night when I stayed out later than the trains ran.
IF I had driven my car downtown to a lot, it would have taken two to three times as long - not to mention finding parking in busy downtown even with parking garages (I know - had a buddy that did that the next year we went), plus the cost of gas and parking for the day (for each day) would have been prohibitive. (We settled on staying at a guest hotel downtown the third year... no driving or trains. yay!)
People in cities with mass transit often prefer it over having a vehicle... and they hate the tourists who bring their cars and don't know how to drive or where to park.
But, back to your point -- you're incorrect. The efficiencies don't take hold when the vast majority of a system is automated -- they take place when only a small fraction is in place. There is a tipping point. If one single car stops to turn left into a parking garage, it can back up an entire left lane of traffic for a mile or more in a decent sized city. That's just one car. For each car that pauses to let someone out rather than turning and seeking parking, you get vast returns in traffic efficiency.
If you must make the public vs private argument, then I'd say you're just arguing quality -- if people care enough, they'll get 2 tiered taxis. One for Uber and another for Super-Uber for those that want to ensure their car is squeaky clean. Most mass transit seats are plastic and easily washable. Cars could easily be outfitted with uncomfortable, but sanitary plastic seating and a bottle of alcohol spray for the germaphobes.
Another aspect is that people junk up their cars with their own crap -- but, it's often stuff they want to keep, so they wouldn't be leaving that in Ubers... they'd just leave trash if they're litter-bugs. I bet Uber could record video and charge extra for damage or littering and put a stop to that (assuming it's paid by credit card).
They key issues for ownership of vehicles are - utility, time, personalization, and storage. People like to keep their baby carriers in the vehicle... sometimes their drinks or other groceries, napkins, kleenex, lotion, sunglasses, etc. Sometimes people store presents in trunks to hide from family members.... various other things.
The personal car isn't going away, but it could become an auto-driving personal car. Still, many families may only need 1 personal family car and use an Uber automated taxi for travelling to work, school, and most other short trips.