It's not a puzzler at all. He knew exactly what he was supposed to do. He needed to be on camera so he had plausible deny-ability to save his own ass.
The television screen in the upper left was running CNN on mute. Noticing the President coming on, Lisa turned on the volume and the crisis conference halted to listen. "...into the World Trade Centre in an apparent attack on our country."
During the pause, I noticed that Brian Stafford, Director of the Secret Service, was now in the room. He pulled me aside. "We gotta get him out of there to somplace safe... and secret. I've stashed FLOTUS". FLOTUS was White House speak for Mrs Bush, First Lady of the United States, now in a heavily guarded, unmarked building in Washington...
"Can you work with Brian", I told [Franklin] Miller. "Figure out where to move the President? He can't come back here until we know what the shit is happening."..."
Against All Enemies
Richard A Clarke
In itself this seems straightforward, but the important part is when it took place: Bush didn't deliver his remarks until 9:30. And yet, the Director of the Secret Service, at least as reported by Clarke, isn't astounded that Bush is still at the school. In fact it seems it's only just occurred to him that something needs to be done. It appears Stafford doesn't want to simply "get Bush out of the school", either -- he wants assistance in planning what their next move should be, from Clarke, who was the man in charge of the White House Situation Room at the time. Which surely was a reasonable idea.