The possibility of switching between atoms does not mean that they are not identical. Even to the contrary: if they were not identical, they couldn't be switched. In my opinion, switching is better viewed as a symmetry.
The fact that the classical atoms are distinguishable is easily remedied in statistical mechanics by including $1/N!$ factor. Yet, it could be an interesting problem to treat nuclei, as if they were quantum particles. E.g., treating a hydrogen moclecule in such a way would require considering the protons on the same footing as the electrons: correctly anti-symmetrizing their spatial and spin wave function components, etc. I doubt that one would obtain new results, but it is certainly a good exercise in quantum mechanics!