Yes you can name a brand name without using their logo.

That's not what the author did here -- he invented a fake logo and chose a domain name and title that include "WaffleHouse." Trademark law is specifically designed to prevent people from creating a fake logo and registering a domain that makes people think it is your own brand. Yes, there is some exception for parodies and criticism, but writing a review / parody sketch is a completely different format than a website that lists information that purports to be official open/closures of your business.

You can't create a product that contains the brand name of another product; I can't invent "Magic Kleenex" or "Better Google," so he similarly can't name something a "Waffle House Index." What he could do is name something "disaster index" or "breakfast restaurant index" where the data happens to be from the waffle house locations.