I draw my answer partially from The Philosophy of Set Theory by Mary Tiles.
PREFACE TO THE QUESTION
The question 'what is the nature of infinity' is an ontological one, because it is the attempt to wrap the mind around the processes of number systems which extend subitization, which is a basic psychological faculty, to number systems which is a theoretical activity. When counting, certain present questions themselves to the naive thinker, such as 'what is the biggest number'. Intuitionally, most physically real things have maxima and minima. There is a tallest family member, or there is a longest finger. It is natural to presume there is a biggest number. Of course, it quickly becomes apparent that that any number given can be incremented by 1. It seems you are asking about what is the essence of infinity? At metaphysical play then is the same ontological muddle that lies behind Tiles question on page 1. "Did Cantor discover... transfinite sets... or did he (with a little help from his friends) create it?" and her statement on page 2. "Cantor's continuum hypothesis cannot be proved from the standardly accepted axioms of set theory."
So, by asking the question, let's note that you set aside finitism as a position. It's only worth noting because some might seek to undermine the meaningfulness of your question by questioning the basis for your question. But let's review cardinality.
In the case of a finite set, cardinality is a bijection between members of the set and a specific member of the naturals, which is closely related to the notion of a sequence which is a correspondence between the order of the elements of the set and the naturals. The question arises, however, is it possible to have a cardinality of a set that is not finite. To this Cantor suggests yes, and offers the notion of using the naturals which are infinite as a measure of other infinite sets by establishing a bijection between their respective members. Thus, the cardinality of an infinite set is a question of whether or not a bijection can exist between the naturals and the set in question. With this bijection as a basis, a new ordering can occur using aleph-nought, and cardinality can be extended beyond a bijection with a specific natural number per se. This is accomplished with the diagonal proof and leads to the the cardinal numbers.
Now, the question of possible and actual infinity should also be reviewed, because unlike cardinal numbers, it is a genuine question in regards to the philosophy of math. From page 25. "Aristotle points out, there is an incompatibility between the notion of a potential infinity and that of a totality... Indeed, there is here the source of another notion of infinity - the absolutely infinite, that than which nothing can be greater." This notion arises from Ancient Greek inquiry into the nature of time, divisibility of matter, and the nature of the universe. Again she cites Aristotle and says on page 26 "The concept of potentially infinity is essentially linked to the idea of a process of construction, of generation, or simply coming to be." She points to the tension between 'universe' which is everything or is the ultimate completeness and infinity which is the sense of introducing new things. On page 27, she goes on to say "If Aristotle is right... if the only viable sense of 'infinite' is that of the potentially infinite, then the universe must either be finite or not a completed whole, not a unity." And further along says on page 28 "comprehensible notions of infinity are intimately bound up with views on the nature of time... between metaphysics of 'being' and that of 'becoming'.
ANSWER TO THE QUESTION
So you say:
*[There seems to be a way to link this idea with the notion of existential quantification re: e.g. Quine's "to be is to be the value of a bound variable," i.e. there is a relationship between the concept "existent" and "having nonzero cardinality." But I'm not sure how to parse this seeming link.]
So the relationship you are seeing in Quine's statement is a metaphysical presumption. That which exists, is that which can be counted, and you are asking after the notion of the existence of things that have transfinite cardinality, that is, cardinality where size is not denoted by bijection with the naturals directly, but by bijections to a bijection with the naturals or reals and so on. Certainly one can accept as first principles Quine's quantification as a condition denoting 'reality' and the transfinites as an extension of a naive cardinality. But instead of using the language that "potential infinity collapses into actual infinity", I'd suggest you say that the existence of 'actual infinity' is a symbolism for the entire process of 'potential infinity' when working in an atemporal formalism.
As far as your statements:
we can preface a variable with one possibility operator first, and then preface that operator by an infinite stream of actuality operators, but this stream will reduce (by the rules of iterated modal operators) to the potential case (and vice versa, if we start from an actual X and then "infer" an infinitely iterated potentiality "behind" it)
Here, as far as I know, you are moving into idiosyncratic or novel formalisms by invoking modal logic notation in regards to statements of infinity and sets. It is standard to assume possibility in quantifier logic with statements like ∃e∀S, where S is an infinite set, but not use modal operators. You certainly could do so, but it would be considered original material as far as I know. If I'm wrong, I'd love to be pointed in the direction of the theory.
So, consider the extended reals. Built into the notion of the interval (-∞,∞) is the idea that if there exists n in the interval, it is actually necessary for any, and indeed every n that there are larger or smaller numbers, not to be named explicitly but constructed by possibility. Hence, '∞' is not a cardinality strictly speaking, but a notation which would translate into the concept of 'possibility'.