This is similar to a question I asked long ago, but there was a misinterpretation. People often say that, for instance, unicorns don't exist, but isn't it more correct to say that there are no unicorns? The first statement ascribes the property of nonexistence to unicorns, which is incorrect, because there are no unicorns in the first place. So, then, why do people continue to make this blunder? This question is slightly different than the previous one because I am not asking if talk of nonexistent objects is a language game.
Imprecise language is getting in the way
You interpret the meaning of "Unicorns don't exist" as "there's this thing called a unicorn which has the property of not existing" which you have properly identified as being incorrect. However, consider this interpretation: "Unicorns don't exist" meaning "The set of all unicorns has no elements", or alternatively "There does not exist an element in the set of all unicorns". Either of these would be equivalent in meaning to your suggested "There are no unicorns" as the definition of "are" in this case means "to exist". I believe this is a case of shorthand being misinterpreted contrary to meaning.