Yes
The testimony of our founders and people, and the history of our nation
It means being a people who have Christianity as our common religious and moral framework
It matters because our system of government is made for that religious and moral frame work and is wholly inadequate to govern any other kind of people
Great quote to this effect from 1816 by John Jay, first USSC chief justice, wrote five federalist papers along with Madison and Hamilton (so I think understands the constitution and intent fairly well)
Providence has given to our people the choice of their rulers, and it is the duty as well as the privilege and interest
of our Christian nation to select and prefer Christians for their rulers.
He also said in the same letter "the gospel not only recognizes the whole moral law, and extends and perfects our knowledge of it, but also enjoins on all mankind the observance of it."
In another letter he said "A proper history of the United States would have much to recommend it: in some respects it would be singular, or unlike all others; it would develop the great plan of Providence, for causing this extensive part of our world to be discovered, and these "uttermost parts of the earth" to be gradually filled with civilized and Christian people and nations. The means or second causes by which this great plan has long been and still is accomplishing, are materials for history, of which the writer ought well to know the use and bearings and proper places. In my opinion, the historian, in the course of the work, is never to lose sight of that great plan."
Or in another place "So far as respects Great Britain and the United States, that obstacle [to the Gospel by treating African heathens as property] has been removed; and other Christian nations have partially followed their example."
As to the question of whether Christians should vote for "infidel" rulers, he cited "Shouldest thou help the ungodly, and love them that hate the Lord?" 2 Chron. 19:2.
He is just one example out of many, but a good one.