Even though the Constitution prohibits any religious test for public office, let's pretend for a moment that it doesn't.
The current President, all nine Supreme Court justices, and 534 out of 535 members of Congress all confess to believing in a Creator. Did belief in a Creator keep them from digging the country into a hole?
If you want to talk about the role of religious philosophy in the founding of the country, then there's no problem with that. But I see no evidence that our current out of control federal government will be tamed if more Americans just start going to church more often.
People create these false either/or dichotomies: Either it is man's doing or God's. I submit that it is both: the former on the micro level, the latter in the big picture. We are getting what we as a people deserve, because all said and done, everyone of those capacities you mentioned, in varying degree of directness, represents us.
I'll ask you a question: Why should the Creator bless us if we as a people spit in His eye?