These Brits, they just don't get it. They cannot understand the significance the Constitution has in the creation and the history of the United States. To them, it's just a written compromise, necessary to somehow bind the newly-independent States together better than the previous constitutional text, the Articles of Confederation, did. And, indeed, it was the product of compromise; but, then, what agreement coming out of a Convention of the delegates of all States wouldn't be? And yes, it did seek to replace the structure of power as it stood under the Articles of Confederation, to a more centralized one; but did that diminish the tensions, the debates, the points raised? The discussion in the State of New York on whether to ratify it produced the Federalist Papers, one of the seminal texts not only of constitutional interpretation, but also of American constitutional theory. The same discussion in Virginia, Massachusetts, New Hampshire, and of course New York, led to the Bill of Rights, the first ten Amendments to the Constitution itself.
Both the discussion in the Constitutional Convention and the one within the States that followed helped to crystallize the views on the role of the Government, which were shared by the Founding Fathers and inspired the revolutionaries. The notion of a government of laws, and not of men; the role of the electorate in determining the course of the government; the notion of enumerated powers for the central authority; of no taxation without representation - all these were written down in the Constitution and these were what would define America as against any other contemporary nation. For America was not a nation based on a common ancestry (though, sadly, discrimination based on ancestry and race found its way into the new Republic through the colonial times), but rather on a shared set of principles and on the ambition to build a "city upon a hill".
It is these sets of principles that bound the nation together - that were the source of its pride - that led to discredited beliefs such as its manifest destiny and to American exceptionalism - that were, also, the reasons behind its greatness. In short, everything that the Americans think that sets them apart from other nations, for better or for worse, is in their Constitution. Anyone who reads the U.S. Constitution will understand that it is, for all the compromises behind it, a Constitution for people confident in themselves and in their abilities; for people, willing to do things themselves, not waiting for handouts or trusting in the charity of the British Crown or any other government. That much of its original constraints on the powers of the federal government have been eroded, thanks in part to the doctrine of the "living constitution", the result being that life in the United States is more regulated today than ever and that the federal government has usurped many of the powers that should have been left to the individual States. This, of course, only adds to the appeal of the Constitution as it was originally conceived, as an instrument that both organizes the government and also sets its limitations (so much so, that arguments against the Bill of Rights included one, according to which the federal government lacked the power to breach those rights anyway). Thence, the Constitution is not a piece of paper to hide behind, but rather it is a text, from which a lot of inspiration can still be drawn.