America hasn't been very British for at least a century, if not longer. I'd say that even the Britain of today is so radically different from the British Empire in its heyday as to be practically a different creature.... thus the above assertions are not, IMO, very accurate.
Many Americans may be Brit-by-blood more or less, but our culture developed amid a very different atmosphere and circumstances. There are fundamental differences that cannot be swept under the rug.
On the contrary, I think literally every major facet of American culture is fundamentally British.
The US speaks English -- from Britain. The US follows the common law -- from Britain. The US political system mirrors the House of Commons and House of Lords, and many political theorists even think the US's system ought more accurately be called Parliamentary than Presidential, because besides Obama, nearly every president has sought the approval of Congress and adhered to Congressional supremacy.
The US venerates English traditions, celebrates British holidays, uses British measurements, and all of its 'formative legends' are fundamentally identical (or outright identical) to those in Britain. Places in the US are named after Britain, and monuments and statues are all in British style.
The US teaches a historical causal chain starting in the same place as Britain's, and following Britain's until 1776, at which point it diverges -- but even then only slightly. Fundamentally, the US has always and continues to look to Europe, Britain in particular, for its 'raison d'etre'.
The US is founded on British ideals of Enlightenment and Protestantism. American democracy? English Civil War/Parliamentary Supremacy. American religion? Thank God for Henry VIII.
The US is built off of British principles of free-trade and entrepreneurism. Industrialism, itself a British invention, was and maybe remains the guiding force in American economic thought.
Even things as minor and minute as the US flag are directly based on Britain, along with things like the US national anthem being in cadence with British marching songs of the time, and the US taking its naming and military conventions directly from Britain.
In sum, the US is thoroughly and totally British -- with the caveat that they don't really like being called such. Yet their language, culture, art, history, government, laws and people are all identical or directly descended from British traditions.
It seems rather obvious from my viewpoint, but I stress that it's not a bad thing. For some reason, some might take this all as an insult -- but it's surely not meant to be. After all, you ARE colonists, how could you possibly not be based off the coloniser? But at the same time you've done something wonderful and exceptional with what Britain has given you, which we all think is very impressive.
Just as a child must acknowledge the parent, don't try to claim the US is not the child of Britain. Instead marvel at what that child has accomplished -- or to quote Isaac Newton: "If I have seen further it is by standing on the shoulders of giants."