First it will be the occasional users or those with a second car or on a budget, but eventually most of us will either pay per use or have an account for regular use rather than outright buying a car like we do now.
For commuting, many will be taking shared vans (possibly pod-based) that pick an efficient route rather than plying a typical bus route.
And we'll get the vehicle we need when we need it. You won't take a ute to dinner with your partner. You won't have a three-door hatch when you need to choose and bring home large items, or transport the family on a road trip.
The future Uber won't even be people you know or don't know. It will be a driverless car.
Self-driving cars-for-rent will get you places both faster and cheaper than possible in your own car. That could make private car ownership seem quaint or even ostentatious in most cities.
Rental cars carry costs that I don't have with my own vehicle: frequent transactions, additional liability issues, wasted mileage driven between fares, uncertain transit times, daily cleaning and inspection for damage, commercial licensing and insurance, middle men, management, marketing, accounting, additional taxes, regulations, and covenants. If I leave something important in my car, it's still there the next day. I don't have to worry about vomit in the backseat. I don't face a transaction cost and a delay (or the uncertainty of a no show) to go to work, to drive home, to hop in the car and go to the mall, or grab a bite. I can leave things in my car. I have less exposure to pathogens and pests from surfaces in revolving contact with thousands of strangers from all over the world, lower probability of exposure to cold and flu viruses, fewer vectors for bedbugs to travel into my home.
I can't see rent seekers (esp short term ones) being so far under my costs that after adding their markup, it will be particularly cheaper for me. The 5 year TCO for a Prius (staple of the ridesharing industry) according to Edmunds is a little over $18 a day at 41 miles per day, 44 cents/mile inclusive of insurance, gas, maintenance, etc. The average fare mile on the Peninsula for a cab is $3. I don't see ridesharing companies finding an order of magnitude in efficiencies and still delivering any kind meaningful profit.
Maybe private toilets will seem quaint and ostentatious, and we'll be soon freed from the tyranny of private bathrooms by happier times of public lavatories.
For city-dwellers, it's different, and the Edmunds TCO leaves out a lot. We pay $hundreds more per month for housing with parking. Then, we pay to find parking near our destinations in extra drive-time, money, and walk-time. And we ride for far fewer than the assumed 41 miles a day, 15K miles a year – so the fixed costs of car ownership are amortized over fewer miles.
That makes the per-mile costs of rideservices already roughly competitive or outright superior for many city-dwellers. That's especially true for the non-poor, who face a higher lost-compensation opportunity costs for every minute spent driving/refueling/parking/walking.
That's also before app-assisted multi-rider pooling, or automated-driving. Those could more-than-halve rideservice costs again.
Parking cost when I'm away go down because the car can park itself somewhere cheaper (in automated facilities), or drive itself home, be at the disposal of friends/family, drop off or pick up things for me. Since in your world their are fewer cars, that puts downward pressure on parking prices. Parking facilities can be located in more economical areas, and pack cars more efficiently because they are automated and instrumented, and their own labor costs are reduced.
One of the main practical use cases for local cabs/ubers is a ride to the airport. Since my car can drive itself home, I really have no need for a ride for hire. Also, since giving a friend a friend a ride to the airport no longer a personal time investment, the odds of getting a ride from a friend increase. If you're worried about me not maximizing my 44 cents per mile, I can take someone leaving the airport home, assuming they meet my standards of reputation. At this point you're probably crowing victory, but note that is purely a highly infrequent, discretionary use of my excess capacity, perhaps less than 1% of miles driven for most people. It's more tax efficient to just swap (robo) rides with friends.
A self driving car will reduce all these onerous costs that ridesharing is supposed to save me from, plus I can earn money from it in your Renters Paradise, so I don't see any downside in owning a car. I hardly feel the cost as it is, even without driving anything close to the most economical, lowest TCO car on the market.
I own my own home, means of transportation, eat most of my meals at my own table...the idea of being a renter of all the major environmental necessities of life is simply not appealing to me in the slightest. I'd rather sleep in my own natural latex, bed bug free bed, and sit on my own sofa or read a book on my own chair, call me crazy. There's never sticky goo on mine, or weird smells. I like private ownership.
For me, Uber is the opposite of "people I know and trust," and I have no desire to have them monetize my movements.