There are (at least) three possible reasons to embark on an axiomatic development of anything:
Such that we can get general results that will transfer to different structures that satisfy the same axioms.
As a way to make establish a convention for what is a valid proof and what is just handwaving. That's important especially in an area where handwaving is known to often lead to false conclusions -- and the history of calculus contains some spectacular examples of smart, capable people being led astray because they didn't yet have solid concepts of what is and isn't a good proof.
Because the axiomatic approach is actually easier than to explain how the things we're speaking about work internally.
The first of these is supremely important in many areas of modern mathematics, but doesn't apply to the reals, full stop. There are no other models of the axioms that are usually presented (up to isomorphism, and as long as we stay within mainstream mathematical foundations, bla bla pedantry yada).
The second one is easy to make idealistic speeches about, and it certainly has something to go for it in a calculus course that will be most students' first encounter with the general level of proof and rigor in it. You can't demand rigorous proofs without first agreeing which facts you can prove things from.
The third one is one that applies very rarely. Usually it's a lot easier to understand a main example or two in at least some detail before moving on to an abstraction. However, the real numbers are arguably one of the few cases where the abstract axiomatic model is actually easier than being more concrete.
In particular, in calculus (and real analysis in general) we often find ourselves needing to prove that such-and-such sequence or function has a limit, according to the particular technical definition of limit. The fact that there is a number that satisfies the convolutions of definition is often not something that's really obvious based on grade-school experience with real-number arithmetic. Our choices are then basically:
Give a complex but complete definition of "what a real number really is", with all the internal details (using, for example, Dedekind cuts), and then prove that based on this definition the limits do indeed exist under such-and-such condition, or
Simply assert as an axiom that real numbers with certain properties (typically something like the existence of suprema) exist, and derive consequences from that.
The first way is, as I've noted, the way that usually works best. But in the particular case of the reals, the concrete definition we start with is going to look very unmotivated to students, and it certainly won't in any way evoke the concept they thought they knew from grade school. In fact, just the opposite: It's far from intuitive that arithmetic on Dedekind cuts works like we expect real arithmetic to work. So doing it the concrete way would actually function as a barrier to understanding.
On the other hand, the supremum property can count as a mostly intuitive fact about how the hand-wavy geometric concept of the real number line behaves. Simply asserting it will allow us move on relatively quickly to interesting uses of reals, while still providing the ingredients for having valid proofs that the limits we need to exist actually do exist.
