My wife and I and an out-of-town guest recently visited the viewing platform overlooking the excavation of the President's House. A U.S. Park Service officer was most helpful in answering our questions and in pointing out the site's significant features.
It seemed to me that the most powerful method of presenting the history here would be to build the interpretive center planned for this location in and not on top of a filled-in site. A glass-walled structure rising from the current dig would fill the center with abundant light. A visitor could stand at one point and view the short distances between the various structures and rooms on this compact estate.
Like Franklin Court and the spare recreation of Benjamin Franklin's house, the President's House and many of its wings and structures could be simply framed to rise above their original foundations. This would enable a visitor to feel what it was like to stand on this very property in the late 1790s, and to imagine what was occurring in its different buildings and rooms.
For their educational value, I would leave in place at least some of the foundations of structures erected over the foundations of the President's House in 1832, and even leave the few city service pipes that currently cross the excavated site. Stairs could be constructed over them.