yes, it is true that there is a "non-standard" model of the reals that is countable. presumably, this is meant to contrast with a "standard" model of uncountably many reals. the trouble is, the "standard model" doesn't exist, at least not in the way people think it does.
what i mean is this: give me an example of a collection of things that satisfy ZFC that includes all sets. i mean, i'd like to know that SOME "version" or "example" of set theory is out there, it would be reassuring. given the group axioms, for example, we can demonstrate that, oh, say the integers under addition satisfy the axioms. it is, to my understanding, an open question whether or not there is a structure, ANY structure, that satisfies the ZFC axioms. at the moment, ZFC appears to describe the class of all sets (let's call this V), and since V is not a set, V is not a model for ZFC (close, but no cigar).
that is to say: it's not logically indefensible to disallow calling any uncountable thing "a set". in this view of things, what cantor's diagonal argument shows is:
there exists collections of things larger than sets (which we certainly know is true anyway).
this is somewhat of a different question than the consistency of the ZFC axioms, although existence of a model would establish its consistency. since great pains have been taken to disallow "inconsistent sets" (the last big push being restricting the axiom of comprehension, for which we previously had great hopes of defining a set purely in terms of its properties), the general consensus is that ZFC is indeed "probably consistent" (it's been some time since anyone has found a "contradictory set").
downward L-S does not show "the real numbers" (with any of the standard constructions) are uncountable, rather, it shows that "the set of real numbers" might not be what we hope it is, in some variant of set theory. indeed the Skolem paradox can be resolved by noting that any model of set theory can describe a larger model, which is what (i believe) current set theory DOES: it shows we can't get by "with only sets", we need a background of things not regulated by the axioms (classes, and larger things).
in other words: there is a deep connection between cardinality, and "set-ness". what cardinals we are willing to accept, determines what things we are willing to call sets. and: what things we are willing to call sets, affects a set's cardinality (cardinality isn't "fixed" under forcing).
1) only finite sets <--> countable universe (first notion of infinity as "beyond measure")
2) countable infinite sets <--> uncountable universe (infinity can now be "completed")
3) uncountably infinite sets <--> strongly inaccessible universe (an infinity beyond all prior infinities)
cantor took step (2) for us, and ever since, we have decided that that pretty much justifies step (3). note that even step (1) is not logically obvious, the axiom of infinity had to be added as an axiom, because we desired the natural numbers to be a set, it does not follow from the other axioms. it is apparently known that (2) is logically consistent, and unknown if (3) is logically consistent (but if (3) is assumed, then (2) follows).
geometrically, the situation seems to be thus: there seems to be a qualitative difference, between "continua" and "discrete approximations of them". the analog and digital worlds are different, although at some levels of resolution, nobody cares.
going back to the real numbers: some mathematicians feel uncomfortable with uncountable sets, including the set (as usually defined) of the real numbers. there are some good philosophical (not mathematical) reasons for feeling this way: most uncountable set elements are "forever beyond our reach", so why use them if we don't need them? perhaps the best answer is that having a wider context (a bigger theory), often makes working in our smaller theory more satisfying: treating "dx" as a hyperreal number, makes proofs about differentiation more intuitive (where we only care about what happens to the "real part").
knowing that sup(A) is a real number, means we can prove things about sets of real numbers in ways that would be difficult, if we had no such assurance. the "background" logic of our set theory (which gets more complicated with uncountable sets) makes the "foreground" logic of the real numbers, easier to swallow.