Sure, that's one kind of restaurant. I've never heard of anyone trying to get a reservation there, though, and I'm pretty sure there's a direct relationship between the lack of demand for reservations, and the restaurant's ability to scale.
If they only had one restaurant because they were incapable of scaling, then perhaps people would get reservations there.... ...but that is not the sort of restaurant they are. They are a restaurant that can scale, and if they only had a single location despite that, they would not have a waitlist.
Or to put it more crudely: If your aunt had balls, she'd be your uncle.
I doubt it; I doubt many would do that at all. In some situations corporate fast-food joints do take reservations (White Castle on Valentine's Day evening is the only example I can think of at the moment) but in those situations they are cashing in on people considering it a novelty. The novelty drives demand to levels that it otherwise never achieves.
If there were only a single Chipotle in NYC, there would almost certainly not be enough demand to require a reservation system (and if they used one anyway, their business would surely suffer). Perhaps you would be willing to call in a reservation for Chipotle, but one person calling in reservations for fast-food isn't going to sustain a business.
Have you had the McRib? It's in this limbo state of being popular enough to warrant introduction into the McDonald's menu periodically, but not popular enough to be a permanent staple.
Basically people will especially go out and get the McRib when it "comes back" and then get tired of it.
If McDonalds was limited to only one location, there would almost certainly be long lines, waitlists, and reservations. The ubiquity of McDonalds, and the fact that they can meet demand enough so that no one needs to wait more than a few minutes contributes to the luxury of not having to wait for a Big Mac.