Again, you are engaged in precious 2nd-guessing of a tedious nature. The WH had been evacuated the morning of 9/11 within minutes, with reason to believe another plane was headed for a crash assault there (Flight 93 which crashed in PA was headed for DC and no one at the time could know what its target might be or if there were more planes and/or other forms of attack underway or imminent).
Playing “Churchill in the Blitz” for the media without knowing the most elementary facts would not help anything. Bush did in fact return to DC by that evening, I believe, but the idea that he should rush there in the first hours is just silly.
I was in the heart of Manhattan on 9/11 (working in a mid-town skyscraper looking out at the WTC burning and collapsing a couple of miles down the island). No one knew then how many more attacks might be imminent on both NYC and DC, or other places, but in those first hours and days it was certainly more important to know that our govt was still functioning to some degree than to believe that the POTUS was rushing into the firing line to share our dangers.
Part of the regular functioning of government is that the capital city is safe enough for the government and its highest officials to reside there. If you do not like the analogy to Churchill during the Blitz, look to Lincoln during the Civil War and his determination to remain in Washington.