This chapter from Shadia B. Drury's Leo Strauss and the American Right is a good, easy read. It is entitled 'American Applications of Straussian Philosophy', and it makes a lot of sense of the different positions, currents, tensions and criticism amongst the American Straussians about the nature of America. I found it very useful as I was having trouble mapping the diverse applications of Strauss in the US.
Shadia identifies and outlines three different American reactions to Strauss's political thought reactions to Strauss with the words denial, despair, and pragmatism.
"The first reaction is a denial that America is modern; a denial that is predicated on a reinterpretation of the American founding as an ancient polity rooted in the great tradition in general, and the classical ideas of the Greeks and the Romans in particular. Harry V. Jaffa is the leading representative of this view. The second reaction begins with an awareness that America is the embodiment of modernity and that her triumph is the death knell of classical wisdom. On this view, the principles on which America is founded are hopelessly modern and tragically flawed. This reaction is understandably filled with despair and foreboding. The gloom of Allan Bloom's The Closing of the American Mind is particularly representative of this view. The third reaction acknowledges the truth of Bloom's vision, but refuses to despair. Instead, it takes a pragmatic approach and sets out to make the most of a bad situation. Willmoore Kendall is the best example of this approach, although it can be argued that Joseph Cropsey, Martin Diamond, and Thomas Pangle provide variations on this theme."
In the light of this diversity of Straussian interpretation I reckon we should stopping talking in terms of a monolithic Straussian position vis-a-vis liberal modernity. Jaffa, for instance, interprets the American constitution in terms of a more conventional understanding of classical philosophy and so breaks with the esoteric writings thesis in which the great philosophers of modernity wrote in a sort of code so as to disguise their real meaning.