Perhaps the way to think of the modality in the Third Way is to think of it not as 'able' or 'possible' but as 'can'. What I mean is this: we can ask of a given thing, given what we know about its duration, "Can it have come to be and can it come to fail to be?" I will only focus on the "Can it have come to be?" (i.e., can it have been generated?) since the Aristotelian position is that anything that can come to be can come to fail to be, i.e., anything generable is corruptible. Also, it will have to be kept in mind that when I say, "come to be" I mean "come to be in the sense that is in view in Aristotle's account of generation". Then we have the following cases:
(1) Something that always exists.
Can it have come to be? No: Since it has always been, there is no point at which it could come to be (be generated). This is necessity in the relevant sense: it cannot be generated or corrupted given that it always is.
(2) Something that never exists.
Can it have come to be? No: Since it never existed, there is no point at which it could come to be (be generated). This is impossibility in the relevant sense: it cannot be generated or corrupted given that it never is.
(3) Something that exists but does not always eixst.
Can it have come to be? Yes: It was not, then it was, there was a point when it could come to be (be generated). It is possible-to-be-and-not-to-be.
Then the basic line of thought in the Third Way would be:
(1) Some things can have come to be (be generated).
: We know this because there are things that came to be.
(2) What can have come to be, at some point was not.
: That is, things that can have come to be cannot always have existed.
: Because if they always existed, there was never a point at which they could come to be.
(3) If the whole world (i.e., everything, taken collectively) can have come to be, at some point nothing existed.
: From (2)
(4) What does not exist can only exist if it is caused by something that exists.
(5) If the whole world (taken all together) can have come to be, nothing exists now.
: From (3) and (4)
(6) Something exists now.
: Look around you.
(7) It is not the case that the whole world (taken all together) can have come to be.
: From (3), (5), (6).
(8) Something or other must have always been.
: Or, to be more precise, something or other exists that has not been generated (=something ingenerable exists).
From this point, of course, we have an ordinary causal argument, on the principle that something that has always been could have always been either because its own nature is such that if it exists it must always be, or because it was caused always to exist by something that has always been.
Incidentally, you'll notice that I keep taking "all beings" and "everything" collectively to mean "the whole world". In this context, I think that is the most natural interpretation: the roots of the Third Way are in Aristotle's De Caelo, and Aquinas very clearly (and plausibly) reads De Caelo as a work of natural philosophy discussing the universe as a whole. Aquinas identifies three subjects of the De Caelo:
(1) the entire corporeal universe, considered as prior to its parts;
(2) simple bodies, considered as prior to mixed bodies;
(3) the first simple body (i.e., the heavens), considered as prior to other simple bodies
If we were to read the first part of the Third Way with purely Aristotelian eyes, it looks like an argument for heavenly body; that is, the part of Aristotelian cosmology it looks most like is Aristotle's argument that the heavenly body is neither generable nor corruptible. Aristotle, as Aquinas interprets him, holds that every form has a power to exist: and everything exists for the extent of time this power to exist covers. The heavenly body, having the most perfect corporeal form, can have no privation of form (it is only capable of privation, and therefore change, of place), and thus always is. Aquinas, of course, doesn't think the heavens have actually existed always; but, as he says, the Catholic view is not that the heavens were generated in the proper Aristotelian sense but that they were caused to exist by the first principle at some point in time. On Aquinas's view, the ingenerable heavens were caused to exist; 'ingenerability' like 'generability' in this context presupposes existence. Given that the heavens exist they cannot have been generated. But although there is no cause of their coming to exist, there still can be a cause of their existence.
However, you will notice a conspicuous lack of any of this in the actual Third Way: the heavens or heavenly body finds no mention at all. This fits with a noticeable pattern throughout the Five Ways: all of them take Aristotelian principles but treat them in a more generalized way than Aristotle himself does. This may have something to do with the fact that they are summaries; it makes sense to put them in the more general form, to avoid potential disputes over details that won't change the main point. In the Third Way it's very likely that Aquinas doesn't want to focus on the heavens alone: another kind of ingenerable is the incorporeal ingenerable (separate substances -- angels, planetary intelligences), and there's no good reason to leave them out in this context. Some have suggested that he's also thinking of matter itself. But strictly, Aquinas doesn't need the specifics: all he needs is that there is something ingenerable because it always exists. This gets him to the point at which he can ask about the reason for its ingenerability; then rejection of infinite regress brings us to a first ingenerable that causes other ingenerables to exist always.