>> Jazz is the only music that’s truly American <<
Jazz combines African-derived rhythms with European harmony. You could say that the combination occurred only here, so that qualifies as “truly American.”
But you can say pretty much the same thing about another quintessential American musical form, the Broadway show. It happened only here, but yet it combined European dramatic and operatic conventions with jazz, ragtime and American “pop music” influences.
Or what about the Delta Blues? The original performers and creators were all African-descended plantation sharecroppers. But their ancestors had left Africa centuries earlier, and their blues music shows almost no trace of African influence except maybe for the use of flatted thirds and sevenths in the melodies.
(Even African-style rhythms are not much used in Delta Blues.)
Finally, what music is more American than the marches of John Phillip Sousa? True enough that his ancestors were Portuguese. But his music? Not American? You gotta be kidding.
You are correct.
There’s a difference between made in America and what the rest of the world thinks ‘sounds American’.