We could say they were or are fundamentally based on British with its Magna Carta and rights of Englishmen with the white Anglo-Saxon protestant morality and work ethic. But it is more than that. It was a spirit of independence where one could rise as far and as high as one's talent's, energy and ability could lead them, differing from their brethren back in the motherland. And not to be overlooked was their rejection of a hereditary based social caste system. People who could not tolerate a master or betters.
This independent attitude and refusal to acknowledge the innate superiority of their rulers would appear to be a problem for an orderly strong man European monarchy. But what, in seventeenth century Britain, is to be done with these malcontents who may desire liberty, but obviously, simply do not want to grovel to a master and obey all his betters tell him just as the majority of the population willingly do. The solution for this sort of independent, go it alone type of weirdo (as any sensible peasant or nobleman would have viewed them) was to have them shipped off to the colonies in the new world and let them hack out towns and cities from the wilderness. It was a good way for the aristocracy to get rid of these malcontents, caused less anxiety for the good peasantry, and put their restless energies to good colonial and mercantilist use.
Then the colonists wanting for complete independence from the mother country was inevitable, finding even minimal obedience and servility to an aristocracy even if they were across an ocean to be extremely distasteful, and the spat over a relatively small 2 percent tax issue and demands for a parliamentary representation was just as good as reason for independence as any. Even if representatives of the colonists and King George III had come to some sort of agreement, the type of people who lived in the colonies likely sooner or later would have pushed for independence and wouldn't have it any other way.
It must be noted that the citizens of the 13 colonies and later the new nation, those who lived in the thirteen colonies prior to July 4, 1776 were not of a single nationality. As well, the revolutionary war was not popular with the majority of colonists, estimated at not quite half, about 40% to 45% supporting (which was enough a critical mass to fight for independence) and a vocal Tory remainer contingent estimated at 20% to 25% with the other 30% to 40% not saying much, going along to get along.
When independence from England was formally recognized after their defeat, the new ethnicity called Americans then formed a system of government according to their nature. Or at least to the nature of the New England descendants of the puritans who weren't not the only Christian sect but also not the only European nationality in the newly formed United States.
When the colonies in the new world declared their independence from England in 1776, the population were mostly English, Irish, Scots or Scotch-Irish, along with Germans at about 30% of the population, Dutch, French with a few freed black slaves, a smattering of Poles and Jews among others. It was not a single ethnicity but a mixed bag of different European nationalities which may or may not include Jews depending on one's view of the subject. Aside from the Germans, I could not find any good demographic numbers for any of the other European ethnic minorities in the mid-18th century. Nor could it be found the percentages of Scotch-Irish and English WASPs.
It should be noted that these different European nationalities were in the past, often rivals or at war with each other. So smoothing away old differences and forming a new national identity and unique culture would take time and isolation to develop. A difficult task considering the colonists couldn't even come to a majority agreement on whether or not to declare independence from the British Empire.
In light of this, it may be legitimately asked exactly what is an ethnic or heritage American beyond anyone alive in the 13 colonies prior to the American Revolution and their descendants? If one were to say a white Anglo-Saxon Protestant, or rather an Englishman, New England puritan or descendant thereof, they may or may not have been a majority of the population in the colonies even though their culture predominated. The Scotch-Irish colonists were Celtic people and although closely aligned with the English, still a different people with their own history and culture and were at times even hostile to the English.
But what of the Germans of Pennsylvania of whom Benjamin Franklin expressed doubts as to their assimilation into or compatibility with the dominant WASP culture and even today as the Pennsylvania Dutch maintain a unique culture of their own? They are, as any proponent of the heritage or ethnic American concept, just as much a heritage or ethnic American as anyone of English descent and whose ancestors fought in the American Revolution and every subsequent war the United States has been involved in. Yet they differ in culture both ancestral and present day. Just as much as anyone of Dutch, Belgian or Huguenot descent.
That is unless a purity spiral were to start among those who would discuss the topic which should include true WASP blood, how many verifiable colonial era ancestors the heritage American can lay claim to and what the minimum number should be for real bragging rights.
In any event, a new country needs a new government. First was the Articles of Confederation intended to protect popular liberty with a weak central government but the founders discovered roughneck individualist types quickly set on each other such that there was political and economic chaos between the states which threatened collapse. Our present constitution ratified on June 21, 1787 was a carefully balanced series of compromises, based on a gentlemen's agreement to obey it. This allowed for a stronger federal government but still respected the American citizen's natural individual rights and property rights as well as an unspoken freedom of association. This emphasis on individual and property rights, Protestant work ethic, and a gentlemen's agreement on the citizen's natural rights that the constitutionally limited federal government would respect would be the core of what could have been the culture of the ethnic Americans. That is, had they been able to fully amalgamate the mix of people already there and develop and clarify this culture without letting in waves of new, more servile peasant-minded immigrants comfortable with traditional old-world, strong man monarchies and later socialism to further muddy the waters.
Here it might be tempting to invoke the nation of immigrants as previously noted and ethnic/cultural homogeneity is important to form a cohesive nation. The now former colonists and newly minted Americans were still mostly united in a more what we could describe today as a classical liberal outlook with libertarian overtones. It was this general view of things, in my opinion, that helped paper over cultural differences making part of this nation based on a general agreement on certain principals and which the sub-cultures are close enough alike to make it all work.
Eventually it would have formed the American which over time should have eroded away or absorbed ethnic and religious differences and amalgamated into an authentic, new heritage or ethnic American had they remained isolated from newcomers. But for a new and growing country in the early to mid nineteenth century with dreams of manifest destiny and an industrial revolution needing more workers, waiting for the ethnic American founding stock to raise enough children on their own was not enough. Immigration was necessary even if the founding stock WASPs found it distasteful but for the some who held sway, politically advantageous and for others, profitable in cheap labor.
If it is true that subsequent waves of different European immigrants did gradually alter the political, social and cultural landscape of America, it is equally true that this was with the general agreement of the ruling WASP Americans and thus the heritage American of that era was complicit in these changes. It was a precedent that like a machine once set in motion began to accelerate along the direction it was initially set on coming to its logical endpoint of the Nation of Immigrants rhetoric, the 1965 Hart-Celler Act ending up with now virtually unrestricted immigration both legal and tacitly accepted illegal immigration. If the heritage American WASP is rightfully unhappy with the current state of affairs, most of the blame for starting it lies with his ancestors.
In conclusion, most of what is an American can be seen in the sense of the heritage American with ancestry going back to the founding ignoring whether those ancestors were English and/or Scotch-Irish or Germans, Dutch, etc. more or less united in a set of preferences and attitudes. This is a function of the culture of European non-preasant mindset outliers and should not be confused with the concept of a nation founded on ideals.
So the ethnic American is not really a nationality in the strict definition of the word as the U.S. was not founded by a single tribe or confederation of closely related tribes creating a country and civilization built on the ruins of previous ancestor civilizations.
So heritage American would be a better term describing present Americans with pre-revolutionary ancestry and with traditional WASP culture.
This is a whole new country carved out of a wilderness, displacing the previous immigrants who settled here from Siberia anywhere from 12,000 to 18,000 years ago. This makes the concept of an ethnic American tricky unless painted with a very broad brush. The composition of the former colonists at the founding used as justification for the present nation of immigrants narrative would be a gross oversimplification and wrong. It must be noted it was mostly immigrants from similar European Christian cultures and most importantly, mostly of people with an individualistic, non-preasant mindset.
Given a thousand or so years, a collapsed civilization or two, taking in almost no new immigrants going forward, a true ethnic American will develop. But for now, it's just too new of a country.