This is an interesting video done, apparently, as an architecture-school exercise in replacing the WH with a new structure. In this case, the structure chosen was a couple of elevator shafts and a staircase, I guess.
This related site is about open-source (brick-and-mortar) architecture but uses a new WH as its central exercise. (Pete, you may be interested in the SketchUp model of the area around the WH.)
This calls to mind a thought experiment. Since the capital was not moved to Cincinnati after the 1814 fire, we have a capital that sits at the extreme east of the nation. Some presidents have maintained a “Little White House” or “Western White House,” primarily as vacation getaways, but do we need a genuine “Western White House” for the president to sometimes work out of and what would it look like?