I don't think the change will be quick, but I think you're overestimating the problems and underestimating the potential benefits here.

Cost is a short term problem. Costs will go down. Liabilities can be insured against.

Here is where I see it going:

Stage 1: Taxis start being replaced by self-driving vehicles at much lower cost. Car rentals too - liabilities potentially go down once they reach a certain level of safety as you don't face the risk of a poorer than average driver. Usage skyrockets as costs drop, and as services can cut pick-up times drastically by more optimally having a larger fleet parked around town and/or driving around town.

Stage 2: We start seeing pooling options from more and more rental providers to deal with high demand situations. E.g. Rental company crunches their numbers and see that my road => the local train station always maxes out capacity during rush hour and decides that rather than buying more vehicles, surge pricing coupled with offering a discount that brings the price back towards normal for each rider as long as it at most takes X minutes extra will be popular and more profitable.

Stage 3: They put in minibuses on some of the most congested streches and/or team up with the local bus companies to launch apps where you can tell them you're at the stop and get guaranteed pickup within Y minutes by either the regularly scheduled bus or a car. You pay a slight premium for the guarantee, which covers the car when the bus won't be there and a profit share with the bus company. (For me the only reason not to consistently use the bus is that if I need to be somewhere urgently, I can't always risk waiting for a bus that might be full; if I had a guarantee that if I press the button and walk to the bus stop, I will get picked up in 5 minutes, it'd make me use the bus more)

Stage 4: As self-drive increase in general, cities put the thumb on parking spots. E.g. in parts of London you already won't get planning consent for housing with more than 1.5 parking spaces per living unit as a means to cap car ownership. Expect to see that gradually driven down, with the expectation that people will buy parking space for their self-driven car elsewhere and/or forgo having one. Driving down the limits on parking will allow for denser developments, making ride share options etc. even more viable.

Stage 5: Youth grow up without depending on their parents to drive them anywhere from the moment they are trusted to go by themselves.

Basically, I see it as a process where the convenience of apps to get you somewhere will keep increasing to the point where people will find themselves increasingly opting to check these apps first and find themselves needing a car less and less. Some transport apps are already combining route-finding with then offering to order an Uber for you.

Expect to see more of that making it less attractive to get a car over time.

Especially as youth get used to a greater flexibility and level of freedom using these type of apps before they can buy a car. Car ownership many places represents freedom from parents driving you around, but more and more teenagers can expect to be in situations were parents opt to order them a car instead of driving themselves.

Before long, a whole generation will experience car ownership as irrelevant to the ability of liberating their transport options from parental control.

Sure, some people will still opt to own one, but many already forgo car ownership, and that number will certainly rise rapidly.