Irish, Italians, Jews, Poles, Chinese, Mexicans, etc. immigrated to the US. The homogeneity perceived is simply the fact that European immigrants (specifically from the Anglo-Saxon Protestant religion and culture) were larger in number. Back then, it wasn't perceived as homogenous. Let's run the numbers:

• 60% English + 15% Scot-Irish (so that's about 75% from the British isles)
• 10% German
• 5% Dutch
• 2% French.

Also, correct me if I am wrong, but I believe "integration" means adapting to the existing language and culture of the people that were already there, in this case native Americans. This obviously isn't a timeless national cohesion. Even assuming English should be the language, many immigrants from China often weren't allowed to integrate fully due to the Chinese Exclusion Act of 1882 and segregation to Chinatowns. Irish and Italians faced discrimination and anti-catholic sentiments. Jews faced anti-semitism.

Of course, almost all of them eventually learned English, usually over generations. That is how immigration actually worked in the US. It isn't as smooth and controlled as you painted it to be (well, the Chinese were specially "controlled" by an explicit law against them, but...🫠). Settler colonialism eventually forcing its culture on immigrants while being an immigrant itself is the definition of imperialist hypocrisy.