The author appears also to misread Paul Atkins, they say:

    Start with the Declaration. Mr. Atkins’s most concrete 
    evidence is Jefferson’s well-worn copy of “The Wealth 
    of Nations.” But the Monticello source cited in his own 
    footnote reports that Jefferson acquired the book while 
    serving in France between 1784 and 1789, at least eight 
    years after he drafted the Declaration. Jefferson may 
    have encountered Smith’s ideas before 1776, but there 
    is no evidence that “The Wealth of Nations” shaped the 
    Declaration.

But in the speech they link to, the words are:

    At the heart of our Founding Fathers’ revolution stood 
    an idea—one that found its fullest expression in two 
    fateful documents, crafted an ocean apart but together 
    in time in 1776: one penned in the summer heat of 
    Philadelphia, the other in the intellectual winds of 
    Edinburgh. 
    ...
    The authors never met, yet their ideas were 
    inseparable, and their contemporaneous works rested on 
    the same conviction: trust the individual, not the 
    institution.

There is no possible way to read that as saying that the Declaration of Independence was shaped by The Wealth of Nations. It pretty clearly states that the "idea" is something that was refined separately but contemporaneously. He also says that the idea was pre-existing, hence that it found it's "fullest expression" in 1776 in the two separate documents.

The closest thing said in the speech that could be read as "the Declaration of Independence was inspired by The Wealth of Nations" is the next section of the speech:

    Thomas Jefferson was no stranger to Adam Smith’s 
    philosophy—he was, in fact, a student of it, evidenced 
    by the well-worn copy of The Wealth of Nations 
    contained in his library, from which he gleaned wisdom 
    that he shared freely with friends and correspondents.
    
    It is little wonder, then, that our founding documents, 
    in many respects, reflect Smith’s central themes that 
    were debated at the time: that government must not grow 
    infinitely.

If you read that quickly I can see how you might read it as "The Wealth of Nations inspired the Declaration" but the reading that I have and that I think was obviously intended was Jefferson was a drawn to the same principles and ideas that Smith was, and so it is unsurprising that these documents written in the same times by likeminded people share common core ideas.

It is really frustrating to me that aggressively hostile readings of things has become such a normal way of writing rebuttals in modern discourse. There is so much to criticize in how the government is run that we don't need to be wasting our time and energy rebutting things that haven't even been said.