Not sure if this is what you are looking for but it is an abstract argument which shows why many of the natural function spaces of bounded functions (continuous, holomorphic, measurable) or operators cannot be separable under the supremum norm.  The reason is that each of these spaces has a weaker, natural topology (often known under the name of the strict topology).  The latter is the natural one for many applications, in particular, for duality theory.  Despite the fact that these do not belong to the standard classes of "nice" locally convex spaces, they do have the property that they satisfy a closed graph theorem for operators with values in a SEPARABLE Banach space (see the first chapter of the monograph "Saks spaces and Applications to Functional Analysis").  Hence if these spaces were separable as Banach spaces, the two structures would coincide, something which only happens in trivial situations.
This argument works in the specific situation that you mention but also on a plethora of related examples.