The common refrain you refer to is taking public in two different meanings. on the one hand, public is addressing all individuals. Then, on the other it is synonymous with government.
This is a problem. An NDO is in fact a discrimination ordinance. It is discrimination against individuals. It is of little difference if your preferences are by choice or innate, This is about group rights over individual rights. I bet you can think of plenty of examples where this kind of thinking has been deleterious not only to those outside of a particular group but to those individuals inside the group.