Using your thinking regarding independence it had been declared prior to 1776 by the actions of the colonies which included forming the Continental Army in June of 1775. July 4 was just the official declaration coming years after the first actions pointing to independence and months after the first acts of armed rebellion and over a year after the founding of the Continental Army.
Colonial legislatures brought a national consciousness to bear and formed a Nation prior to 7/4/76. That consciousness predated states and produced such declarations as the "Declaration of Causes and Necessity of Taking Up Arms" and later directed the states to write constitutions. They all realized that only by uniting could they hope to suceed in defying Great Britain.
When are you ever going to tumble to the fact that what you are repeating is just a lot of authoritarian, antirepublican propaganda uttered in order to overawe the People and the States into obedience to a self-promoting business class? When are you ever going to get it?
This "Mystical Union" stuff would put me to sleep, if I weren't aware how obnoxious it is, and how nefarious the intentions of the people promoting it.
Anyway, the foundation of the current Government and social compact rests on the Constitution, not the Declaration or any Revolutionary document. The Treaty of Paris has more legal significance for the United States than the documents you're arguing over. Even the Declaration of Independence is a scoping document and a statement of causes and mission, not a social compact or the foundation of a government -- or the Union.
The term "American" as applied to British colonists in North America, had been in use for some time. It was often used by the British in a derisive sense. You might find amusing the Swift essay, "A Modest Proposal" (circa 1727) and its characterization of Irish and Americans. He uses both term with a decided nationalistic sense.