Simple answer: Because there is no museum to house it.
Bull. There's a number of museums in DC's Smithsonian complex. The collection could be rotated through there, a new set every month. It would probably draw lots of regular visitors.
The Army doesn't want to, because they want their own little empire.
Raising the funds to build the museum has been the main problem. The majority percentage of the cost must be provided by the private sector, and the donations have not been coming in. The Army is providing the land at Ft. Belvoir, provided the good citizens of Virginia don’t begin another “not in my back yard” protest that has cause 3 relocations of the proposed spot in the last 10-15 years.
The totally private Army Historical Foundation is the fund raising organization for the National Museum of the US Army. A museum for the Army was authorized by Congress in the 1800’s but the cavaet with that no govenment funds would be spent.
Being somewhat aquainted with the Smithsonian, their exhibit schedule is based on 6 and 12 month rotations, not monthly.
Also, don’t worry the Center of Military history is having its staff cut by 25% as part of the DoD headquarters reduction over the next two years.