[Edit: After digesting the comments, doing more research & more thinking, I have reformulated my question]
On the iterative conception of sets, as it is usually expressed, there is no "completed" $V$, but rather an unending series of stages which are built up iteratively in a process which can never be completed. The obvious objection is that an abstract platonic object is not something which can be "built", nor can it appear in "stages" if it is timeless. -Source
I was previously under the impression that $V=\bigcup\limits_\alpha V_\alpha$, however this comment states that this "should really be thought of as a theorem". Another comment gives the formulation $V=\{x:x=x\}$. The latter formulation seems to incapsulate what I was orginally trying to ask about (a type of "completed" "timeless" universe which is not constructed "from below").
My primary question: does there exist a top-down formulation of set-theory?
My secondary question: would it be accurate to say that the cumulative hierarchy is a (inner?) model of $V$ (eg. $\bigcup\limits_\alpha V_\alpha \in V$)?
The usual way to think about set theory is that there is some universe $V$ of sets that satisfies some axioms. So $V=\{x:x=x\}$ is the correct definition of $V$ as a class. Assuming a strong enough fragment of ZF (e.g. including power set and some replacement), we can define the cumuluative hierarchy $V_\alpha$ by transfinite recursion. And then, if we also assume the axiom of foundation, we can prove that $V=\bigcup_\alpha V_\alpha.$ To emphasize that this is a theorem and not a definition, I'll note that this equation is really short-hand for the theorem $$ \forall x \exists \alpha\; x\in V_\alpha.$$
If we don't have foundation, then we cannot prove this. In this case, we do not have $V=\bigcup_\alpha V_\alpha,$ but $\bigcup_\alpha V_\alpha$ still makes sense as a class. This is called the class $\mathrm{WF}$ of well-founded sets, and if foundation fails it just means that there are sets that are not in this class, i.e $\mathrm{WF}\subsetneq V.$
It is accurate to think of $\mathrm{WF}$ as an inner model of $V$. It is a model of ZF that is a definable submodel of any model of ZF - foundation. (However it is not accurate to say $\mathrm{WF}\in V,$ as you have put it. $\mathrm{WF}$ is a proper class, so it is not an element of $V.$ Perhaps what you meant was $\mathrm{WF}\subsetneq V,$ as I wrote above... which is only true if foundation fails.)
As to what this means about there being a 'top-down approach to set theory', I'm not completely sure. Based on what I gather you mean, the answer is yes, as I've described, what you heard in the comments is the standard mathematical way of thinking about things.
However if we're taking a more philosophical angle, we might not be satisfied with this... do we really have a sharp conception of or motivation for this universe of sets that we think of ZFC as describing? We might find it useful to think about it as being built in stages, at least as a motivating principle. One difficulty here, is we don't really have a good answer for what the "full" power-set of a set is or what "all" the ordinals are... so the principle with which we are constructing it with are vague. It's not any less vague than just saying "we have some universe of sets", but it's not obvious it adds anything other than intuition. On the other hand, there have been philosophically motivated attempts to develop set theory in a way that puts the cumulative structure more at the foreground and less as a consequence... it's just not the way it's typically developed and taught.