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.