It seems you are confusing two separate ways to define the real numbers.
First, there's the axiomatic approach, where we define a set of axioms (of the real numbers) that the real numbers should satisfy, and work with the object defined by these axioms.
Second, there's the constructive approach, where we work in a stronger theory, like ZFC, to construct an object that we then declare to be the real numbers (i.e., we construct natural numbers, then rationals, and define the reals as (equivalence classes of) limits of Cauchy sequences).
However, it is easy to get confused here, since we talk about the axioms of ZFC in the constructive approach as well as about the axioms of the real numbers in the axiomatic approach: the term "axiom" is used in a different way between these to: in the constructive approach the axioms tell us what construction steps are allowed, while in the axiomatic appreach the axioms tell us what properties the reals should satisfy. To avoid confusion, I will use "axiom" to refer to the axioms of ZFC, and "property" to refer to the axioms of the real numbers.
It is a bit of a perspective issue, see also my answer on this question.
We don't use the property of completeness (of the real numbers) in the construction of the real numbers; it is a property that the reals should satisfy, but not a tool that helps us build them. Instead, we construct the real numbers with Cauchy sequences, and then we prove using the axioms of ZFC that these constructed real numbers satisfy the property of completeness (and all other properties of the real numbers).
What this shows, is that the constructed real numbers form a model of the theory that is given by the properties of the real numbers. There exist several models for the real numbers; instead of using Cauchy sequences, we could define the reals using Dedekind cuts, or as a subset of the Surreal numbers. However, the nice thing about the properties of the real numbers, is that we can prove (in ZFC) that any two sets that satisfy the properties of the real numbers are isomorphic to each other. Therefore, up to isomorphism, we will always construct the same object.
This is different from, for example, the properties of a field, since we can construct several fields that are not isomorphic to each other (the rationals are not isomorphic to the reals, for example). The proper term for this is to say that the properties of the real numbers are categorical. We can therefore talk about THE real numbers, instead of SOME (set of) real numbers, since the method of construction does not really matter (up to isomorphism).