A service will be able to keep cars near you, and will have sufficiently higher utilisation of their cars to be willing to pay more for parking spots. Your waiting time to get a taxi service will likely drop substantially at the same time as your waiting time to get picked up by a car you own is likely to go up.

As someone who doesn't own a car, the only potential appeal to me of owning a car is shorter waits and predictability (always there). If the predictabiity and waits drop for rental services, my reasons for considering buying a car would rapidly drop. If the waits to use a car I own go up, my reasons for considering buying one would drop further.