The United States was founded to be a small l liberal small d democracy. All small d democracy means is voting (i.e., consent of the governed). And small l liberal means certain rights are antecedent to majority rule. The Declaration of Independence is the quintessential "liberal democratic" document. The US Constitution established the United States as a constitutional republic. And a constitutional republic is simply a form of "liberal democracy."
You do see quotations from the Founding Fathers criticizing the concept of "democracy" in favor of "republicanism"; but what they are really criticizing is direct democracy or mob rule. There are certain "republican" checks that must be built in to the democratic rule of the people. Chiefly, it's representatives not "the people" who make the laws. Another republican check is that certain individual rights are prior to majority rule. Ultimately, all of the following terms correctly describe the American system of government: "liberal democracy," "representative democracy," "representative republic" or "democratic republic." The following term does NOT: "direct democracy." And that's what the Founders criticized when they noted our form of government was a "republic" not a "democracy."
Now, on to David Barton's distortions of this dynamic. This should help explain why so many Christian Nationalists repeat "we are a republic, not a democracy" as a mindless mantra:
If you study the history of "republicanism", you see it was invented by Western Culture's noble pagan Greco-Roman heritage. The Greeks tried direct democracy, saw that it didn't work. And the Romans pioneered "republicanism." Indeed, the Founding Fathers looked back at noble pagan republican Rome with a great affinity. Indeed, when they wrote with surnames they picked ones from republican Rome, not the Bible. The Founders thought of themselves as new versions of Cato, Cincinnatus, Brutus, Novanglus, and of course Publius. Yet, what they were arguing for wasn't exactly what ancient Rome had, but rather something more modern. 18th Century republicanism was more of an Enlightenment construct.
The Bible spoke little to ideals of 18th Century republicanism. The authors of the Federalist Papers never quoted the Bible in support of the provisions in the US Constitution. But outside of the Federalist Papers, some Enlightenment thinkers and ministers did "read in" republicanism to the Biblical record. As rationalists, America's Founders and the ministers who worked with them picked and chose from all sources of world history, including the biblical record, what they thought "rational" and ignored or discarded the rest. And along the way they did a lot of "reading in" to those sources what they wanted to see to fit their "Whig" ideology.
When framing the Constitution, Noah Webster perfectly captured this Enlightenment zeitgeist that undergirded the US Constitution:
Of all the memorable eras that have marked the progress of men from the savage state to the refinements of luxury, that which has combined them into society, under a wise system of government, and given form to a nation, has ever been recorded and celebrated as the most important. Legislators have ever been deemed the greatest benefactors of mankind—respected when living, and often deified after their death. Hence the fame of Fohi and Confucius—of Moses, Solon and Lycurgus—of Romulus and Numa—of Alfred, Peter the Great, and Mango Capac; whose names will be celebrated through all ages, for framing and improving constitutions of government, which introduced order into society and secured the benefits of law to millions of the human race.
This western world now beholds an era important beyond conception, and which posterity will number with the age of Czar of Muscovy, and with the promulgation of the Jewish laws at Mount Sinai. The names of those men who have digested a system of constitutions for the American empire, will be enrolled with those of Zamolxis and Odin, and celebrated by posterity with the honors which less enlightened nations have paid to the fabled demi-gods of antiquity.
But the origin of the AMERICAN REPUBLIC is distinguished by peculiar circumstances. Other nations have been driven together by fear and necessity—the governments have generally been the result of a single man’s observations; or the offspring of particular interests. IN the formation of our constitution, the wisdom of all ages is collected—the legislators of antiquity are consulted—as well as the opinions and interests of the millions who are concerned. In short, in it an empire of reason.
Now, Webster wrote these words in 1787; later when the French revolted and tried to construct their "empire" on "reason," it didn't work so well and Webster, apparently, rethought his confidence in "reason" as the rock that undergirds the American republic. In the 19th Century Webster began talking like a Christian Nationalist, and consequently, offers Barton et al. many quotations. Here is the offending article from Wallbuilders.
The quotations criticizing democracy from the other Founders are apt; but, as we have seen, what they criticize is direct democracy. But what Barton quotes from Webster for how a republic defines is an extremely self-serving and distortionist definition. Here is what he reproduces from Webster:
[O]ur citizens should early understand that the genuine source of correct republican principles is the Bible, particularly the New Testament, or the Christian religion.13
The other quotations are from Founders explaining how they believed "law" ultimately traced to the divine. The Founders did believe in an immutable God given "natural law" discovered by reason. But they disagreed on the proper relationship between reason and revelation. The most notable "republicans" like Jefferson, J. Adams, Franklin and Madison were rationalists who believed man's reason penultimate for discovering that "higher law," in both private and public life. But I'm sure many other FFs disagreed. Because they disagreed on the Bible's proper role in politics (how to properly understand the Bible led to sectarian arguments which they were trying to avoid), they formed a consensus that, in politics, we would look to "reason" to determine God given natural law.
Now, that there is an immutable "higher law" that trumps majority rule (indeed it's the source of "unalienable rights") was one republican check on democratic rule. However, it's arguably not the sine qua non of "republicanism." If anything the central feature of republicanism v. direct democracy is representatives make the law, not "the people."
But to David Barton and those who follow him, the difference between "republicanism" and "democracy" is democracy is majority rule, republicanism is "God's law." And of course "God's law" simply reduces to the Bible to be used as a "proof-texting" trump as today's evangelicals currently do. So when Ben Franklin said America was given “A Republic, if you can keep it,” he apparently meant, according to Barton, that we must act like evangelical proof texters believing the Bible the infallible Word of God. And, by the way, Ben Franklin believed "that the[re are] several Things in the old Testament impossible to be given by divine Inspiration,..."
Barton's theory ignores the fact that the Founders turned chiefly to "reason" and avoided proof texting the Bible to determine the content of the God given natural law. As John Adams put it:
To him who believes in the Existence and Attributes physical and moral of a God, there can be no obscurity or perplexity in defining the Law of Nature to be his wise benign and all powerful Will, discovered by Reason.
– John Adams to Thomas Boylston Adams, March 19, 1794. Adams Papers (microfilm), reel 377, Library of Congress. Seen in James H. Hutson’s, “The Founders on Religion,” p. 132.