the belt drive is not too different from those used in cars. They don't stretch like tires, because they have a fabric belt that's injected with rubber.
The only problem with belt drive is the lack of length adjustability, which makes it a hassle to get the right gear combination. With chains, you can just cut them to whatever length is required.
As noted, modern drive belts don't stretch so that isn't a problem. They do have a couple of issues though.
First, they can't be separated for installation like a chain so the frame has to be specifically built with a separating drive side dropout so the belt can be installed or replaced. You can't use a belt drive in a standard frame without a lot of modifications.
Second, you do have to be able to adjust belt tension so horizontal dropouts of one form or another (like a fixie or IGH) are required.