By "perturb the existing surface normal", I think what you mean is that we use normal maps defined in tangent space, so that when the normal map is applied it acts as a displacement (loosely speaking) to the underlying geometric normal of that surface.
One reason to do this is simply that a tiling texture can be designed, where the texture can be applied to surfaces of any orientation. For example, a single brick texture could be used on walls, a floor, or a ceiling. All those surfaces have different geometric normals, but we can re-use the same normal map for all of them by using their tangent space as a basis for applying the normal map.
A similar issue affects animated characters, where for instance you want to move the character's arms and legs around using an animation at runtime. Tangent-space normal maps enable the fine texture detail to follow along with any motion, without needing to alter the contents of the texture.
If I am to overwrite normals (that is, replacing the normal on the surface with the corresponding one from the normal map), would I not have more 'control'?
You as the author of the normal map would have more control in some sense, but do you want that control? It's usually more useful to create normal maps that can adjust to the shape of the mesh they're applied to. Otherwise, every time you change the shape of a mesh, you would need to update the normal map to match. And if you got it wrong, the lighting on the model would just look completely messed up. The information provided by the underlying surface geometric normals is useful - you don't want to just throw that away.
It's true that normal map strength can also be readily adjusted in tangent space, by lerping the normal map toward or away from (0, 0, 1), which is the default or "identity" normal in tangent space. This is more of a side benefit to the tangent space technique than its primary purpose. Other texture transformations, such as rotating or blending between different texture layers, can also be done readily in tangent space.
Another side benefit is that tangent space normal maps are easier to compress without losing too much quality, since they have effectively fewer degrees of freedom and a narrower typical range of values.