Isn’t saying that the concept of existence is flawed because it fails to distinguish between existents a bit like saying the concept of ice cream is flawed because it doesn’t distinguish between vanilla and chocolate?
I would agree with the above in the following way:
The subject-object distinction is still possible in Objectivism–indeed, Piekoff fleshes it out a bit in OPAR. The mistake that your correspondent made is that he seems to regard the subject, or predicate, as a kind of independent Platonic being. A good example of the subject-object distinction is any predicative assertion, say, “The ball is red.” The object is the ball, which is subject to the description “red” (Note: I’m not sure that the reason the distinction is named “subject-object” because the object is “subjected to” the predicate. I just choose this way of writing because it flows and helps the explanation.). In this example, your friend would be asserting that there is some kind of thing that is “redness” that is not accounted for in the axiom of existence–only the object, the ball, is accounted for.
This could lead to a whole conversation about Platonism, which has its place, but not necessarily when discussing the axiom of existence. Case in point, the axiom of existence, were it accepted by a Platonist, could also make a claim on the existence of Platonic forms.
So the axiom is not challenged at all, but rather the question is raised, “What, exactly, exists?” Or perhaps otherwise put, “Can anything exist that is not extended in space or endures through time?”
Interestingly enough, though, I had a grad student friend at Duquesnes who wrote a fairly persuasive argument that the subject-object distinction can be eliminated by set-theory notation. E.g. {the Eiffel Tower} = {made for the Universal Exhibition} (intersection) {in Paris} (intersection) {tall}. I would argue that, though it’s now hidden in the grammar, the idea or the form of the Eiffel Tower is manifest, which is to say the subject has an object, and so the distinction is merely painted over.