Plane travel isn't a day-to-day event, even for those wealthy enough to afford private jets. You still have to pack, travel to an airport, get on the plane, crew has to be made ready, flight plan has to be filed, etc. It's a high-impact event that occurs at most every few days, even for frequent flyers.
Driving is a day-to-day task. My time's already short, if I have to add an extra 5 minutes waiting for pickup every time I want to go anywhere (once again an optimistic estimate, it would likely be closer to 10), on a busy day that's a minimum extra hour a day of time I've lost just standing on the sidewalk. Margin matters, even an extra 15 minutes a day adds up to over 90 hours a year.
Also these rental services/configurations will hardly be free. So the cost of the car/maintenance isn't eliminated entirely. For my situation, given that I've been driving the same car for the last 11 years, and it was ~16k when I bought it, plus maybe an average $1200 a year in insurance, and maybe an average of $300 a year in maintenance that means I've spent an average of $2,954.54/year over the last 11 years on car ownership. And that number is only going to get smaller unless I buy another car.
But in this scenario I'm losing a minimum 90 hours per year on the margin. If those were work hours, then at my current take-home rate I can decrease that number down to around $1065/year going forward. Given that I'm likely going to make more money as the years go by, these continuous rental services are going to have to be awfully cheap to be worth it for my situation, and I don't think I'm too far from the average.
Now sure, if you're buying a new car every 4 years like some people do then it might be worth it. Or if the cost of insurance is prohibitively high in your given location. Or if your job sucks and you just can't afford a car. Or if you actually need a broad variety of vehicles on a regular basis. There are lots of factors that go into it, but for anyone who's middle class or better I don't think it'll supplant ownership entirely. Or maybe it will and I'll just be with the sour-grapes number-crunchers in the corner ranting about margin to anyone who'll listen. :)