I have seen non-American "integration" of foreigners, in Europe, Middle East, Asia, etc. In these places there are much more distinct mono-cultures relating to race than there are in the US. What is the dominant culture in NYC? Yuppie white Midwestern transplants? Orthodox Jews? Blue collar Italian and Irish descendents? American blacks? Mexican immigrants? All of these groups have very different standards for demeanor and behavior. In this way, these groups of "New Yorkers" collectively permit "assimilation" because there isn't really anything in particular to assimilate into. As long as you follow the law and spend a lot of money, people typically won't be that unkind to you or treat you in a way you'd recognize as a "foreigner", because they already treat everyone as a bit of a "foreigner".
I suspect we're disagreeing over what it means to "assimilate" or "be accepted as" one of an ethnic group. Imagine a stirring speech where a speaker says "I have dreamed of a respite from the trials our people have undergone". When the audience hears "our people", they are going to think of a certain group that they imagine themselves belonging to. In America, a miniscule fraction of the population thinks of "everyone who is legally an American" when they hear the phrase "our people".
However, some other place like Paris, the massive city-state of the white Franks, has a carefully guarded culture that they gatekeep as a majority ethnic group. You can live in France, and you can speak French, but you can't "become French" because many of the people who call themselves "French" think specifically of white Frankish/Gaulish/Roman descendents. That's just not happening in major American cities.