The thesis of the article seems America was Cicero's country not Aristotle's. America's Founding had a strong "Greco-Roman" component (however they "re envisioned" that heritage). And yes, if you draw a distinction between the Greeks and the Romans, it was the latter who more influenced the American Founding than the former.
The author is also aware that Thomas Jefferson, in "his famous letter of 1825 to Henry Lee," claims Aristotle as one of the four principle sources of the Declaration along with "Cicero, Locke and Sidney.”
But then, like a scholar with a thesis, Birzer explains away the import of that quotation.
(It's possible, as Birzer notes, to draw a distinction between the "Founding" or "Foundings" as represented by the Declaration and by the "Constitution." John Locke, for instance, profoundly influenced the Declaration in the sense that Jefferson quoted part of Locke's Second Treatise on Government and the Patriotic Preachers likewise quoted Locke for the principles of revolution in the face of Romans 13; but Locke's influence on the Constitution is debatable. Perhaps Aristotle was like Locke in this sense.)
Here is nice passage from Birzer's article:
When James Wilson, one of only six men to sign the Declaration as well as the Constitution, and a future member of the U.S. Supreme Court, gave his famous lectures at what is now the University of Pennsylvania in 1790 and 1791, describing the meaning and philosophy of the American founding, he offered an almost purely Ciceronian vision of Natural Law and Natural Rights. Though he draws upon Aristotle here or there, he constantly refers back to Cicero, though his Cicero is, admittedly, more mythologized than real. As with John Adams, the two revered Cicero, focusing almost exclusively on the Roman’s Stoic ethics.Note the lead to Birzer quoting George Washington's first inaugural address:
"When Washington famously submitted the following on April 30, 1789, he did so much more as a Roman than a Greek:
"There is no truth more thoroughly established than that there exists in the economy and course of nature an indissoluble union between virtue and happiness; between duty and advantage; between the genuine maxims of an honest and magnanimous policy and the solid rewards of public prosperity and felicity; since we ought to be no less persuaded that the propitious smiles of Heaven can never be expected on a nation that disregards the eternal rules of order and right which Heaven itself has ordained; and since the preservation of the sacred fire of liberty and the destiny of the republican model of government are justly considered, perhaps, as deeply, as finally, staked on the experiment entrusted to the hands of the American people. (First inaugural address)"This article, to me, smacks of the logical fallacy of the "false-dichotomy"; yes Washington thought of himself more as a Roman Statesman than a Greek; but did not one system of thought lead to the other? And how is Aristotle incompatible or in any way not complementary to Cicero?
Likewise the notion that there is an "indissoluble" connection between virtue and true happiness is, as far as I understand him, Aristotle's Ethics 101. (Groundhog Day was a wonderful representation of that teaching.)