A conceptual answer can be given in the framework of this answer.
The functor that sends a measurable space X to the set of random variables on X, i.e., equivalence classes of (unbounded) real or complex valued functions on X, sends colimits into limits and satisfies the solution set condition. Hence it is representable by the representable functor theorem.
The representing object is a (complete) measurable space Z such that morphisms of measurable spaces from X to Z are exactly random variables on X.
The measurable space Z has very interesting structure. For example, it contains a copy of real (or complex) numbers with the usual Lebesgue measurable structure. It also contains an atom corresponding to each real number, a copy of Cantor set with non-Lebesgue sets of measure 0 etc.
The underlying reason of these effects is that preimages of sets of measure 0 under morphisms of measurable spaces are again sets of measure 0. Thus a morphism from an atomless measurable space X to real numbers equipped with the usual Lebesgue structure cannot be constant, because a single-point set has measure 0 in the reals. Thus we have to add an atom for each real number and a lot of other stuff to get all random variables.
In fact, one can/should view the process described above as the canonical functor F from the appropriate category of topological spaces to the category of measurable spaces.
This functor should be contrasted with another canonical functor G from the category of smooth manifolds and submersions to the category of measurable spaces.
Every manifold is a topological space, however, G does not factor through F. One really needs the additional structure of smooth manifold to define G.
However, there is always a canonical map (actually, a monomorphism) G(M)→F(M) for every smooth manifold M.
Real (or complex) numbers form both a topological space and a smooth manifold, thus we get two measurable spaces F(R) and G(R) out of them by applying F and G respectively.
It is the former measurable space that should be used in the definition of random variables, not the latter one. In other words, we should think of R (or C) in the definition of random variable as a topological space, not as a smooth manifold.
Thus there is no reason to deal with non-complete spaces. Complete spaces are indeed technically superior to non-complete spaces.
Unfortunately, the problem stems from the fact that measure theory was born probabilists seem to be unfamiliar with this relatively easy construction and is still perceived as a part instead they have to phrase their definitions of analysis. Combining this with the fact random variable in a way that other mathematicians are unwilling to write textbooks relies on an area traditionally regarded as a part of analysis, we should not be surprised with the current very sad (from the viewpoint of category theory) state of measure theorysubtle difference between Borel and Lebesgue measurable sets.