"Black swans do not exist" isn't wholly a statement about everything that exists. It's a statement about the possibility of the property of "being a black swan" to exist.

It is a statement about an abstraction, not existence. Like saying "the cup is above the table" is a statement about the existence of the "above-ness with respect to the table" of the cup. One needn't survey all of reality to know if it is possible for some abstraction to exist. It need only be consistent with the rest of your language, so that's all you need to survey.