Why can't we just add "nothing else is a set" as an axiom?

It's not clear why we need to. One reason we "can't" has to do with a technicality. ZF is a theory in "first-order logic". Saying that nothing is a set except that which the axioms imply is a set is not a "first-order" statement, so if we tried to add it we'd no longer have a first-order theory. That would be too bad, because first-order logic works out a lot nicer than higher-order logics.

There's no way to say "nothing is a set except things that the axioms imply are sets" using just $\forall$, $\exists$ and $\in$.

(In fact, the axioms don't quite say what you say they say! There's no mention of "is a set" in the actual axioms. What you say is maybe how one thinks off what the axioms mean, but in fact, for example the actual axiom (2) is $$\forall x\forall y\exists z(\forall t(t\in z\iff (t= x\lor t= y))).$$The axioms talk about sets, not about what is or is not a set; adding "is a set" to the formalism makes it a totally different sort of thing.)

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Set Theory

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