I think what you mean when you said that $x \in A$ must be a logical ``function'', is that it is an assignment that sends a pair of sets to a truth value, of course each object in each pair is a set that can substitute the symbol $x$ or the symbol $A$, in this sense $x \in A$ is called a "propositional function", you can refer to Russell on this in his "History of mathematical philosophy". Your question is legitimate since in order to know a function, then its domain must be specified in order to complete the characterization of a function, now its range is known which in binary logic it is $\{T,F\}$. So the domain can be seen as a set of all $sets$ that the axioms are speaking about, notice that the circularity is only apparent, i.e. if you think that the domain of discourse must include ALL sets as elements of it, then clearly the domain of discourse cannot be a set, and you'll be into searching for this "weaker" notion that you mentioned. But that's not how things are understood, the understanding is that the elements of the domain of discourse are the sets that we are speaking about by our axioms and this doesn't include the domain itself. If you want you can add a primitive constant symbol $V$ and relativize all axioms to this constant [i.e. all quantifiers are written bounded in $V$]. So the theory is not aiming to speak about all sets, it is only aiming to speak about sets within $V$, more specifically it only speaks about sets that have the characteristics that are specified by the axioms, not of every possible set. Given this partial sectoral understanding, the apparent circularity would vanish. Of course I'm speaking in relation to $\text{ZF}$ and related extensions. On the other hand there are indeed theories that includes the universe of all sets spoken about by the theory among the objects its speaking about, $\text{NFU}$ would be such an example, but here the circularity is obvious and actually admitted. But in the context of $\text{ZF}$ set theories, nothing of that is endeavored, so you can keep having stronger and stronger extensions with each extension defining the universe of discourse of the lower theory, and you can go along that indefinitely, and again without being involved in any circular issue.
If you are not content with this and want some other kind of 'collection' other than sets and classes, then you can go to Mereological totalities, perhaps those would prove to be weaker than sets in your sense. So you can refer to work on "Mereology" which is about Part/Whole relation.
