EDIT: OK, the following is a bit long. The tl;dr is the following:
If we're skeptical of philosophical commitments such as Platonism (which I think we should be), then the right response to the circularity involved in defining mathematical objects in terms of sets while recognizing sets as mathematical objects is this: that all semantic reasoning, such as the development of model theory, is really syntactic reasoning taking place in a formal theory which we're choosing to interpret as being "about" objects whose existence is dubious, false, or meaningless. These syntactic claims (such as "ZFC proves that no set contains itself") are just statements about finite strings, and we can make sense of them even in a purely empirical way.
OK, now the long version:
Based on your edit (as far as I can tell, your "implementations" are just models), I think you're asking:
To what extent do we need to make set-theoretic commitments to do model theory?
(Note that I said "model theory," not "logic;" I'll say more about that in a moment.)
The answer is that we do in fact need to presuppose a notion of set. If one is a Platonist, this isn't necessarily problematic, and a formalist will dispense with the entire apparatus altogether and simply look at the formal system it takes place in (again, more on that in a moment).
There is also the option that what we really have here is a way of taking any "notion-of-set" and producing a corresponding model theory; this is exemplified by topos theory, where each topos can be understood as a universe of sets and model theory can be developed inside the topos. Based on your most recent comment to me, I think this might be interesting to you, but ultimately it runs into the same problem: we wind up having to talk about some sort of mathematical objects to develop semantics for mathematical statements, and this is ultimately no less circular or demanding of Platonism.
Now, what if we are unwilling to make any set-theoretic commitments at all? One approach is to argue that the whole semantic apparatus of model theory, and indeed all of mathematics, is not describing anything but rather is simply taking place inside a formal theory. That is, we don't view the statement "If there is a countable transitive model of ZFC, then there is a countable transitive model of ZFC + CH" as really referring to "countable transitive models," but rather is simply a string of symbols which has been produced by a certain formal system. The fundamental question of formalism, to my mind, is why the formal systems we do math in are valuable and interesting, but there's no doubt that formalism provides a vehicle for doing mathematics with the minimal philosophical commitment.
Now, after all, we do need some commitments to get off the ground. For "naive" formalism, this amounts to a commitment to the "existence" of the natural numbers in some sense; further examining this notion, we can try to reduce the philosophical commitment involved even further. For example, "truly empirical" mathematics is extremely ultrafinitist: the only things one is allowed to assert is "the string $\sigma$ is deducible from the strings $\sigma_1, ...,\sigma_n$," and only in the case when one actually has a formal deduction of $\sigma$ from $\sigma_1,...,\sigma_n$.
Why am I bringing this up? Well, the point I want to make is that formalism helps us not worry (as much) about circularity without invoking some kind of Platonism. Specifically, while one can be suspicious of set-theoretic foundations of mathematics because of the circularity involved in defining mathematical objects via sets while sets themselves are mathematical objects, a claim like "ZFC proves $\sigma$" is universally intelligible. Essentially, what this means to me is that we can do mathematics as if we were making Platonists without actually making the philosophical commitments involved in any serious way, and still be doing "honest mathematics" - the point being that the formalist perspective gives us a bulwark to "fall back to."
This "optional Platonism," I think, is why mathematicians tend not to care about these issues; we tend to recognize that we could reduce all our reasoning to concrete statements about finite strings, and therefore that our Platonist statements can be translated into obviously meaningful ones.
Of course, this translates (one of) the Platonist challenge(s) - "In what sense can mathematical objects be said to exist, and why are we justified in claiming that they do?" - into the "formalist challenge:"
What criteria determine whether a formal theory is "mathematically valuable"?
I have strong and wrong opinions on this matter, but I think that's off-topic for this specific question.
