That is exactly it.
I admit I do not know how debated this is among philosophers, but in mathematics, such formalizations of quantification over sets is ubiquitous and non-controversial. That’s not to say that nobody ever made any attempts to formalize things differently since the advent of Zermelo’s and Fraenkel’s set theory and the subsequent reconstruction and axiomatization of mathematics. But suffice it to say that these days, pretty much all of the maths any of us learned at school, as well as all of the higher maths used in engineering and the sciences is rooted in a formal logic that defines its terms in just this way.
A unicorn does not have horn, if no unicorns exist to say this about. But “all unicorns have a horn” is not a statement about if or how many individual unicorns exist. It is rather “For all x: If x is a unicorn, then x has a horn”. It doesn’t matter how many objects x that happen to be unicorns there are. It only matters that of all the unicorns that there are, not a single one is without horn. If there happen to not be any unicorns at all, that’s fine, too: It is even then still the case that there are zero hornless unicorns.
Let Charlie be a unicorn, and let’s suppose Charlie the unicorn exists. In that case, between the propositions “Charlie the unicorn has a horn” and “It is not the case that Charlie the unicorn has a horn”, exactly one is true, and the other is false. These propositions are a true dichotomy. Moreover, the truth-value of the proposition “Charlie the unicorn has no horn” may well depend on the truth-values of the ones above. We are naturally inclined to say that it cannot have the same truth-value as “Charlie the unicorn has a horn”.
Now suppose Charlie the unicorn does not exist, and let’s go with your suggestion of properties being only true for existing things. In that case “Charlie the unicorn has a horn” is false, for the having-a-horn attribute can only be assigned to things that exist. However, “Charlie the unicorn has no horn” is also false, because, again, having-no-horn is also an attribute, and those can only be assigned to things that exist.
So knowing the lack of some attributes of an entity is insufficient to conclude anything about whether it has any others. We have to also know whether the entity exists, only then can we begin doing logic to statements about it. Most troublingly, contradictory attributes (or lacks of such, as the case may be) are outright allowed for entities that happen to not exist.
Now, I’m not saying this is necessarily a bad thing. It is possible to keep this from exploding into total contradiction-spraying absurdity, if we just introduce some rules of inference that restrict what can or cannot be derived from propositions, depending on whether things referred within them exist or not. All I’m pointing out here is that there may be intuitions on this topic that find the implications of your suggestion concerning, and that a legitimate case can be made against adopting it, just as intuitively as your case in favour of it may be.