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"But during Bush’s first 8 months, he was getting slammed in the press (notwithstanding he’d gotten a solid tax cut enacted and No Child Left Behind approved by both houses in Congress) and those brief remarks at the 9/11 site were so genuine and heartfelt that everyone realized we had a leader on our hands who was not going to whimper away with his tail between his legs but instead stand up to the perpetrators and ensure they paid a heavy price for their perfidy."

I think you've got the right idea, but the wrong event. The entire nation pulled together in the days after 9/11, and GWB acquired their support with the speech he gave about a week after 9/11. Recall the comment that followed by Pat Buchanan: "Tonight, George Bush became President".

From the September 21, 2001 speech:

"These terrorists kill not merely to end lives, but to disrupt and end a way of life. With every atrocity, they hope that America grows fearful, retreating from the world and forsaking our friends. They stand against us, because we stand in their way."

"We are not deceived by their pretenses to piety. We have seen their kind before. They are the heirs of all the murderous ideologies of the 20th century. By sacrificing human life to serve their radical visions - by abandoning every value except the will to power - they follow in the path of fascism, and Nazism, and totalitarianism. And they will follow that path all the way, to where it ends: in history's unmarked grave of discarded lies."

http://www.guardian.co.uk/world/2001/sep/21/september11.usa13

Bush’s first comments to the nation on the day of 9-11 were viewed as “shaky,” as if he weren’t quite in command of the situation or how to respond.

The Ground Zero remarks with firefighters were on 9-14.
http://www.americanrhetoric.com/speeches/gwbush911groundzerobullhorn.htm
Bush seem much more in command while also exhibiting the kind of empathy that apparently Americans had come to expect from a president under Bill Clinton. I think the key was this was a largely unscripted moment in which Americans saw their president as both a “man of the people” who understood and really felt the loss of the families affected by 9-11, but also as a tough no-nonsense commander in chief who was resolved to strike back at the perpetrators as opposed to getting the UN to pass a resolution of protest against the attack. People could see he was speaking from the heart—not reading from a canned speech or teleprompter. The crowd reaction said it all.

I concur his subsequent Oval Office speech likewise carried through on this “strong but compassionate” leader theme. But I’m reasonably certain I’m not alone in believing 9/14 was the first day this view of Bush had begun to emerge.