Unless they can make money or somehow keep them dependant, they'll try. Most likely, you're correct. But, if there is a way to give them a 'free' car, they'll surely look into it.
You may live in or near a city that had "free bicycles" to encourage people not to drive their car.
A person could pick a bicycle up at one spot then ride it to where they wanted to go and leave it for the next person.
They were painted horrible fluorescent colors so no one would steal them.
Guess what happened next?
Now if they give away cars, what do you think will happen?