I am sure you have thought about these things more deeply than me. Some of it, however is mere semantics: naive set theory (as far as I understand the history) led to some rather problemmatic paradoxes, which were remedied in stages by modifying the axiom system in use (proposed?).
As I see it, the axiom of infinity puts "limiting behavior" into set theory...the most striking, and perhaps important, example of this is "proof by induction" (which apparently has the same strength as the axiom of infinity). I agree that the original poster has some confusion about enumeration, which is quite common: people think infinite numbers or objects should behave like finite ones, only "bigger", but this is not so. The "acceptance" of "completed infinities" into mathematics was somewhat of a begrudging one...in fact, "proof by induction" is only valid because we deem it so, we cannot derive it from the other properties of ZFC, nor point to the natural world for empirical validity.
I certainly did not mean to imply ZFC asserts the existence of inacessable cardinals, just that the same "extension process" from which we pass from the finite to the infinite can be "turned again on itself" (repeatedly). That is, we can form a heirarchy of cumulative hierarchies (of which ZFC forms only the first), and then once again (and again....). I believe at some point we run out of names for these strange beasts.
As I understand it (poorly, at best) forcing is used to show "equi-consistency" of ZFC+other stuff with ZFC. Certainly you are correct that these methods are not needed to "conceive" of these notions...that was very poorly worded on my part...I just wanted to express the idea that lots of stuff lies beyond the purview of ZFC, even if we can express these notions in set-theoretic terms.
I recall reading in a category theory text once that there would be no need (in that text) for anything beyond "ensembles" (the ensemble of all (proper) classes being an example). It is heartening to see such unabashed optimism amid such uncertainty (I said ironically).