I dunno, I think Korngold was about as American as Rachmaninov was (though granted the later wrote very little after he immigrated). Or maybe Stravinsky is a better example.
Yes, perhaps Stravinsky would be a good example. Also Dvorak, who wrote his 9th and his Cello Concerto in the U.S., but did not establish roots in the U.S.
As for Rachmaninoff, after immigrating to the U.S., he wrote his revision to Piano Concerto No. 3, composed Piano Concerto No. 4, Symphony No. 3, Rhapsody on a Theme by Paganini, and his Symphonic Dances -- the latter two being among his best and most performed works. Rachmaninoff lived all of his adult life in the U.S. became a citizen, and is buried in the U.S. alongside his wife. I don't think most people would refer to him as American despite all that, but the music he wrote while living in the U.S. could go in this folder -- except that he's not unsung, of course.
A much more substantial case can be made regarding Rudolf Friml. He moved permanently to the U.S. in 1906 at age 17 and settled in as a happy philandering Hollywood composer for more than 60 years.
To my knowledge, Korngold never expressed much interest in returning to Europe, although I've not read any detailed biographies. And the majority of his most substantial and successful works were composed in the U.S.
The problem is, the U.S. is a country of immigrants. Everyone here, except for Native Americans, are from other parts of the world. So, one hears a lot of African American, Italian American, Asian American, Mexican American, Irish American, etc, etc. Those labels adhere even if one is 3rd, 5th or 10th generation in this country. People take pride in their heritage, yet still consider themselves Americans.
At what point does one's country of birth become less relevant than one's place of life and death? Yo-Yo Ma was born in France of Chinese parents, but has lived in the U.S. since his mid-teens. Most everyone considers him American. Stokowski was born in London, with an English father of Polish heritage and an Irish mother, but he moved to the U.S. when young and became a citizen, so most people consider him American. Friml moved to the U.S. at around the same age as Yo-Yo Ma and Stokowski. Why would he not be considered American as well?
I don't have clear cut answers for many cases. Most people in the U.S. accept as their own anyone who has permanent residence in the U.S. and becomes a U.S. citizen. Heritage is appreciated, but they are Americans.
Of course, others can disagree. That's just my opinion.
And my apology for getting off topic. I'll add some more American music to counter this tedium.