The same problem happens with car sharing: If you have the car in your driveway, using it is trivial. If you add booking a rental/shared car, going to the station, checking it for damage, ... a quick trip turns into a major chore.
And even if it may be overall cheaper to rent as needed than owning the "overkill" solution the whole time, it won't be much cheaper, because renting is ridiculously expensive. Short time rentals and car sharing around here are typically $50-100 per day and/or not that expensive per time but $0.50 per km which makes it prohibitive for longer trips. You can get an OK used car for a couple thousand and it will cost you way less than that in maintenance.
Self-delivering self-driving cars could alleviate at least the hassle (since you'd be able to reserve a fungible car from a citywide pool, for pickup and return at your door), although the cost aspect will likely remain.