This has been on the back of my mind for one whole semester now. Its possible that in my stupidity I am missing out on something simple. But, here goes:
Let $M$ be a topological manifold. Now, even though $C^\infty$-compatibility of charts is not transitive, it is true that if two charts glue with all the charts of a given atlas, then they are compatible with each other.
Given this, one may conceivable define an equivalence relation on the atlases of a manifold, and then consider the equivalence classes. But, we do not do this. Instead we define a differentiable structure to be the maximal atlas (which being uniquely built up from a given atlas will be in its equivalence class).
Why do we do this (apart from its arguable simplicity)? Why don't we take the equivalence class of atlases in stead to be the differentiable structure?