Who created the country, and its institutions? Whose language and legal system do we use? Who set the tone for what was to be assimilated to?

Other than language (which ignores other languages coming in and going out), these aren't consistent. Tons of other ethnic groups, for example, have made several contributions to what is considered "American" culture. Just because there was originally an English cultural hegemony in the founding of the United States does not negate the diverse origins and current diversity that helps characterize it.

It wasn't the Indians, it wasn't the Cajuns, it wasn't the Delaware Swedes, it wasn't the black or Irish slaves, it was the English through and through.

Who owned the resources of the country before English settlers took and re-appropriated them? Who served in the economies that built the institutions? Contrary to what you may believe, America has not been a stagnant nation since 1776. And by the way, a specific founder of a civilization does not mean "ethnicity"; Mughal India did not mean "ethnically Indian" (which has problems on its own) was solely appropriated to the Mughals?

Up to the 1960's, Native Americans referred to those people.

This is wrong historically; the term "Native Americans" is surely from the 1960s, but it should be known that the general term for "American" was always in a flux. Originally perhaps limited to the Anglo Saxons, but over time incorporating other European groups such as the Germans and Irish, and eventually incorporating groups like Asians.

Again, your reductionist history does no one favors. If we are saying that the English had a big impact on the founding of this country, or that American culture has a strong Anglo-Saxon basis, I'll agree. But in terms of ethnicity as we define the word, there really is no such thing as ethnically American.