The answer to this is a firm yes. Rumsfeld was widely known to be not of this century in his thinking, and Cheney wasn't much better.
While Bush himself was definitely a moral absolutist, which is maybe not ideal for high office, he was a lot more clever than he got credit for and was a lot more sympathetic to the disadvantaged than many in his party, making his ideas of compassionate conservatism that he espoused in his campaign and his tenure as Texas Governor likely a lot more prevalent in his administration.
Bush's Born again Christian outlook is vital to understanding his ideas about both the world and his place in it. While many people were not fans of this, had this outlook shone through during his presidency more, he likely would have been a centrist economic president rather than trying to squeeze the dot com bubble for all it was worth. And foreign policy wise, there is no way to know how he would see things.
There is no way that Bush does not expand our defense commitments in Eastern Europe. That was integral to his world view. If 9/11 goes forth as normal, taking out the Taliban and Al Qaeda also definitely happens. The question is on Iraq. I don't know how that would go, but Saddam radicalizing at the pace that he was might make war more or less inevitable there by 2006 if Bush is still around. Iraq was becoming a terror state, Saddam was no longer the secular asshole dictator of the 90s but rather a Sunni Islamist asshole dictator.