This is not quite right - a specification is not equivalent to writing software, and the code generator is not just a compiler - in fact, generating implementations from specifications is a pretty active area of research (a simpler problem is the problem of generating a configuration that satisfies some specification, "configuration synthesis").
In general, implementations can be vastly more complicated than even a complicated spec (e.g. by having to deal with real-world network failures, etc.), whereas a spec needs only to describe the expected behavior.
In this context, this is actually super useful, since defining the problem (writing a spec) is usually easier than solving the problem (writing an implementation); it's not just translating (compiling), and the engineer is now thinking at a higher level of abstraction (what do I want it to do vs. how do I do it).