How to think like a set (or a model) theorist. Kenneth Kunen in his “The Foundations of Mathematics” writes:


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*‘Set theory is the study of models of ZFC’ (p. 7) 

*‘Set theory is the theory of everything’ (p. 14)


With (1) Kunen is pointing to a change in the intended use of the axioms of ZFC: ‘there are two different uses of the word “axioms”: as “statements of faith” and as “definitional axioms”.’ (p. 6).
With (2) he means ‘set theory is all-important. That is


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*All abstract mathematical concepts are set-theoretic.

*All concrete mathematical objects are specific sets.’ (p. 14)


According to (1), to be a set is to be any of the individuals of the universe of a particular model of ZFC, just like being a numeral (standard or not) is being any of the individuals of the universe of a particular model of PA (here I am using Shoenfield’s terminology in “Mathematical Logic”, p. 18).
But, according to (2), models are sets too, as any other objects dealt with in the metatheory.
What's more, models of set theory are defined in terms of relative interpretations of set theory into itself, a syntactical concept. (See Kunen’s “Set Theory. An Introduction to Independence Proofs”, p. 141), which makes the whole thing a bit more confusing.
The view of the axioms of set theory as "definitional axioms" is appealing. And more in regard of (2) since then they pretend to define all that there is. The study of models of set theory has an intrinsic interest, but why reduce the study of set theory to it? Or stated another way, why abandoning the old view?
I would like to know if set theorists do stick to one view or another or shift comfortably between both at need, and the reasons they have to do so.
 A: I highly recommend reading Andrej Bauer's excellent answer to an earlier question along the same lines. To summarize the situation in a few words: the fact that set theory is anterior to model theory does not preclude the model theoretic study of set theoretic structures. The reason for that is the same as why we can study the ring Z even though integers are conceptually anterior to abstract algebra.

I will attempt to answer your revised question as I understand it.
First, it is important to note that set theorists only rarely work with general models of ZFC, so it would be unfair to reduce set theory to the model theory of ZFC. Indeed, set theorists almost exclusively work with well-founded models of (fragments of) ZFC. In fact, the models that set theorists consider are usually of the form (M,∈) where M is a set and ∈ is the true membership relation. While model theoretic ideas play a very important role in set theory, this is very different from how a model theorist would approach models of a given theory.
Second, the language commonly used by set theorists is designed to accomodate multiple philosophical views and focus on the mathematical ideas. When dealing with forcing, set theorist often use a lingo popularized by Baumgartner. In this lingo, one talks of the generic extension V[G] as an object of the same level and equally tangible as the set theoretic universe V. This is easily translated in the multiverse view, but it is just as easy to think of this as no more than a linguistic convenience: set theorists who prefer working with countable transitive models à la Kunen can mentally replace V by such a model M and interpret the words as those of an inhabitant of M; set theorists who prefer working with boolean valued models à la Jech can simply interpret the generic filter G as a convenient abstraction; set theorists who prefer the formal approach can translate the lingo into purely syntactic arguments; etc. In my case, none of these translations ever take place, but I do hold the possibilities in a safe place for comfort.
In conclusion, my advice is to focus on the mathematics and use models as tools just like any other mathematical object.
A: I think the more fundamental question to ask is why set theorists insist that the axioms of set theory be strictly first-order in nature (*).  I claim that you can't really explain the motivations of set theorists until you address this phenomenon.
In light of this near-universal assumption, I think it is fair to say that:


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*First-order logic is the foundation of set theory (which is in turn the foundation of other things)

*The axioms selected for set theory are those which enable as much model theory as possible, without risking inconsistency.
The fact that one can then turn around and use model theory to study set theory itself is a major bonus, but not really a part of the foundations argument.  I've heard some quasi-mystical tales about sets existing in some alternate universe out there, but I feel far more comfortable using "it lets me do model theory and hasn't led to contradictions" as a justification for the axioms.
(*) Excluding, for example, second-order quantification, the logic $L_{\omega_1,\omega}$ of countable conjunctions, or the "exists uncountably many" quantifier.
