Earlier quoted context omitted.
It's not about the internet not working in general. These routers are about dead spots in places that have functional internet already. It's a pretty common problem that absolutely does exist.
I live in downtown Manhattan and have that problem, but like OP asked, I don't think it's a huge enough problem to warrant a separate router just for the purpose. Most people simply use the router rented out from their internet provider (for free), and in most cases just make it work somehow (instead of buying a new device just for routing they would rather acknowledge that wifi doesn't work well in some parts of the…
OTOH, for lots of people the cost of a new router is a small fraction of even a years ISP service, so if it significantly improves performance of the internet service you are paying for monthly, and is usable for several years, its worth it.