That said, only a percentage of people would want to just discard the car they've spent considerable money on.
Self-driving fares can never be as cheap as regular driving seems. With regular driving, you are silently depreciating the value of your vehicle whereas a self-driving car would have to include depreciation in each ride.
However, for people who do (or plan to) use bicycles or mass transit a significant proportion of the time, being able to fall back on an hailed autonomous car in case of rain, haste, whatever is very appealing. It just depends on being able to look at a monthly bill of $200 (or whatever) in on-demand rides and see it as a huge step forward from what you used to pay in all the fixed costs of car ownership whether you used it or not.
The real trick will be having a workable solution for families with kids— that's an area where current taxis/rideshare have a gap, since they're not willing to supply their own carseats for liability reasons.
But when you have that, you double or multiply by N, the road usage. Which is going to produce multiple problems for everyone.
People imagine that once autonomous vehicles reach a certain level, they'll just morph to provide all the benefits of a well-planned public transit system. But I'm doubting common good can just fall out of more chaos.
And hey, the private automobile itself was thrown onto society with effectively no consideration of externalities and brought us to our current heaven/hell of urbanism. So it's quite possible that, despite some talk of planning, autonomous vehicles will just appear, make a hash of things and all the clean-up will happen later, if at all.
The carseat thing is a real missing piece in the puzzle though.
If I own/rent the car, I can just leave them in while I'm doing whatever. If I'm doing Lyft, I'm stuck carrying around baby seats between rides. Which is not appealing.