Periodically, one reads of an evangelical leader or Republican legislator who believes that the Bible has a great deal to say about America. Yet biblical scholars are buzz-killingly insistent that all of the biblical writings were composed during a time when no one in the Eastern Hemisphere had any idea that the Americas existed. Even more buzz-killing: if there is an analogue for America in the Bible, surely it is Babylon or Rome, both of which are demonized as simultaneously opponents and unwitting tools of God.
There is an existing version of Christianity that gives evangelicals everything they want: Mormonism. As people my age learned repeatedly from watching commercials offering free Books of Mormon, that book includes accounts of “other sheep” who will “hear my voice” (John 10:16) — i.e., Jesus’s post-resurrection visit to America. It’s a bold retcon, but it’s not the only one. It makes the American West the explicit promised land. It dials the emphasis on family up to 11 compared with traditional Christianity. It takes a belt-and-suspenders approach to textual inerrancy: the original document is written on metal plates (hence no need for a manuscript tradition that might introduce errors), and Joseph Smith’s translation is “re-inspired.” And if there is any worry about the Church becoming irrelevant or behind the times, there’s a principle of progressive revelation that takes the fundamentalist idea of dispensationalism (where God has different requirements in different historical periods) and shifts it into the contemporary world.
Perhaps there are signs of a rapprochement in the evangelical embrace of Romney last time around and the general trend of greater alliances on social issues. But if anything, the question is why it’s taken so long — Mormonism actually is what evangelical Christians think Christianity should be.
(Note: I do not say this to make fun of Mormonism, which I think is a really interesting historical phenomenon and which, all things being equal, seems to contain a similar mix of bad and good as evangelicalism or any other mainstream contemporary religious movement.)
An und für sich 
“If mankind were capable of deriving the most obvious lessons from the facts before them, in opposition to their preconceived opinions, Mormonism would be to them one of the most highly instructive phenomena of the present age. Here we have a new religion, laying claim to revelation and miraculous powers, forming within a few years a whole nation of proselytes, with adherents scattered all over the earth, in an age of boundless publicity, and in the face of a hostile world. And the author of all this, in no way imposing or even respectable by his moral qualities, but, before he became a prophet, a known cheat and liar. And with this example before them, people can still think the success of Christianity in an age of credulity and with neither newspapers nor public discussion a proof of its divine origin!”
John Stuart Mill, Diary entry, 10 April 1854.
I think the one flaw in your reasoning here (and otherwise the whole post strikes me as eminently reasonable) is that Morminism is a traditionalist, backward-looking religion; it construes America into theology, but a 19th-century cover-waggons big families version of America. 21st-century America is centrally about several things that Mormonism isn’t very well placed to apprehend, like celebrity, a nation in therapy (using ‘science’ to purge the buried traumas: everybody has a buried trauma), about science-fiction and the future. That’s why I’d replace Mormonism in your argument with what seems to me the quintessential 21st-century American faith — Scientology.