12th Chief Directorate originated from the Main Directorate of special weaponry of USSR MoD. This organisation was created in 1947 to coordinate R&D, production and storage of nuclear weapons and to develop guidelines on their usage (and on defensive measures against similar enemy weapons). As a result, by 1951, when training started for actual military personnel who would be using the nukes (the Soviet Air Force), the Main Directorate of special weaponry already had a sizeable infrastructure dedicated to transport, storage and maintenance of the weapons - note that these storage sites were also responsible for the final assembly of nukes.
In addition to storage bases, in their R&D capacity, the Main Directorate was responsible for operating the testing grounds and nuclear detonation detection facilities. When SORT treaties were made, they also were charged with operating decommissioning facilites, too. They are still responsible for long-range nuclear detonation detection laboratories (and control over international treaties implementation, but I guess these guys don't have too much work on their hands these days).
This gave the organisation a very particular skillset. They basically control the most of Russian nuclear weapons' lifecycle (which is why they are so secretive); RVSN (and other potential users) only receive fully ready warheads, they are responsible for putting them on delivery systems and priming, but nothing else - they used to have some storage bases, giving them that overlap you mentioned, but these were transferred to the 12th Directorate when it was formed.