Why don't Americans claim English ancestry?

This is a complex question. I will provide you with a very generalized macro answer of one aspect of how this phenomenon came to be, but realize there are many factors in play, and that this is not a unique situation to America.

The United States has experienced several notable immigration waves, the bulk of each comprised of individuals from particular regions or nations. In other words, large numbers of people from the same cultural background coming all at once.

At the onset of the American Revolution, there was not such marked ethnic lines in the Colonies like we have in America today. An analogous identifying factor, though, was whether one considered oneself a "Patriot" or a "Loyalist." In the mind of a Loyalist, they were first and foremost British citizens. A Patriot, too, might just as well feel a strong connection to their European heitage, but to them, one's identity as something "not-" or "no-longer-British" became paramount. Overall, the Patriot sentiment would eventually win out, facilitating the diminishment of British-ness as a self-identifier. But either way, ethnic background insofar as "race" was not a divisive element, because there were no strong divisions thereof.

Subsequently, once an "American" identity had solidified, large immigrations of ethnic groups inevitably caused friction: A native New Yorker in 1820, whose New World roots go back 75 years, might not take much notice of a Irish family moving into the neighborhood. But when literally thousands of Irish move into the neighborhood, bringing their own culture and values with them, suddenly he might feel like his home is being infringed upon. And now you have racial prejudice.

When one's values feel threatened, there's push-back: people naturally cling to their traditions and cultural identities, whether we're talking about one person or a million people. But when the numbers are so large, it's easier to stand firm in your identity and resist assimilating too swiftly into the surrounding culture. As a result, notions of ethnic differences have more opportunity to become more firmly entrenched. The sheer magnitude of immigration has given America it's unique character, and has also allowed for multiple senses of pride in one's heritage.

And therein lies the difference: Before this phenomenon could manifest, a nacient American identity formed out of, and replaced, the British one, arguably because the opposite conditions existed at the time.

The first United States census began on August 2, 1790. Unlike modern census questions, the early versions usually were limited to 6 questions:

name of the head of the family and the number of persons in each household of the following descriptions: Free White males of 16 years and upward (to assess the country’s industrial and military potential), free White males under 16 years, free White females, all other free persons (by sex and color), and slaves.

Place of birth was added much later, when the USA came to be a "melting pot" of immigrants. Ethnic purity was maintained for a time, but the heart wants what the heart wants and ancestry became convoluted, even for early British Americans. Distant ancestry becomes a vague memory.

Families have seemed to claim recent patriarchal ancestry. My personal example is that I grew up thinking I was Sicilian since that is what my grandfather was and my father claimed. However, I am only 1/4 Sicilian from my father and 1/2 Norwegian from my mother. So as a "white" person, which would I claim, not to mention the plethora of other nationalities thrown in the mix from my paternal grandmother who was Scots-Irish (whatever that involves). So, for me, my claim is American.