I genuinely don’t see why a brand name can’t be the proper name of a point of interest.
If a Burger King restaurant presents itself as simply Burger King, and that’s how people refer to it in conversation, for me its name is Burger King. That’s how people tell it apart from other businesses on the street including the KFC next door, consistent with the proposal: “Proper names name a single object in a group of similar objects.”
I agree that if a restaurant is publicized as Burger King X, where X is some kind of geographical or other identifier, that may be a more appropriate name. But around me, they don’t do that. If a distinctions needs to be made between two Burger Kings it would be done with the street address or the name of the enclosing shopping centre if there is one.
Having said all that, none of this really seems relevant to simply restructuring the page. As suggested by @Wynndale, it could be avoided (or deferred to a future discussion) by simply not saying anything about using brands as names.