FDR or LBJ could make it if they want as part of the New Deal or Great Society. I can't see anyone else doing it.
I could see the National Archive, the National Film Registry, the national libraries, National Endowment of the Arts, and the national monuments under this department. The WPA under the New Deal had a large art program that funded painters, book writers, play writers, sculptures, etc. Perhaps that could survive until the modern day.
The department, however, would not have much real power. Despite that, the Republicans, would constantly pledge to get rid of it but they never would.
And power of political correctness? First Amendment blocks that shit right out.