This assumes the occasional need for a car, because even with the best public transit etc. in the world, there will be cases where not using a car is impractical.

If you need a car on at least a weekly basis, you're probably going to have your own car either way (unless the self driving car services are really good and cheap).

But even if all everyday trips don't require a car, it's very likely there will be some exceptions. And those can make or break this. If getting a car for that occasion requires hours of overhead (e.g. getting to a pick up/drop off point), is sufficiently inflexible (cars not reliably available on short notice), or prohibitively expensive (e.g. per-km charges on car sharing cars that make a couple longer day trips per year more expensive than just getting a cheap car), people who otherwise could do without a car will consider getting one.

OTOH, if the alternative is really good, people who occasionally need a car might use a service rather than owning a car, which means usage-based cost i.e. a much bigger incentive to pick alternatives. If they have been pushed to own a car, the fixed costs are a sunk cost and the marginal cost of taking car can easily be cheaper than public transit.