Most nations of the old world have had people living in its constituent lands for several centuries at the least. Immigration has not meaningfully shaped them,

Most nations of the old world have had people living in its constituent lands for several centuries at the least. Immigration has not meaningfully shaped them, unless we talk about specific instances of forced population movement (partition of nations, ethnic cleansing etc.). The US has for most of its history had a significant population of immigrants, rising up to 10% of the population at the turn of the 20th century. It has absolutely been made by immigrants; people who decided to move to the US in search of better opportunities.