True - but at least shows that there is a possibility for immigration from the Ottoman Empire at the very least. Also, just kind of a fascinating story!
We just can't underestimate how much the American Government went to bat for Chistian missionaries in the late 19th century. A war scare nearly erupted with the Ottoman Empire in the 1890s, created by the various murders of American missionaries and various massacres of the native Christians (even the Armenians). It never amounted to anything, as the US never had sufficient naval strength to actual support the missionaries in Anatolia and Syria (a far cry from Tripoli, so far from Constantinople).
So, even if there were a wave of migrants, it likely wouldn't be in large enough quantities, as by the late 19th century the US was already throttling immigration into the country. If any were allowed, it might be increased Armenian/Assyrian/Chaldean/etc that manage to make the trip west, as they would be the ones sponsored by missionaries for immigration west.
Not that it's necessarily impossible, but difficult to do given a similar US compared to OTL.
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One far-fetched idea might be Ablania, in fairly modern times. One very far fetched idea, as I have found out previously that it was mostly a small movement based on hearsay and a few articles in various papers. There was no organized movement to actively achieve it. But, say, a more interventionist and internationalist USA could lead towards it, especially if there is no stigma against overseas states (likely rising from overseas states made prior). Or, instead, it could become an alternate version of a protectorate create in the aftermath of a dissolution of the Ottoman Empire (to keep the Italians and Austrians away from Albania, even). Over time, Albania remains close to the US, with the populace eventually voting to become a commonwealth a la Puerto Rico. And, eventually, this movement translates into statehood.
Say, standard Wilsonian Armenia remains a US protectorate with a more internationalist USA, while there is a nominal protectorate over Albania (based upon OTL sponsorship of Albanian independence). Albania requests formal protection thanks to Italian and Austrian/Greco-Yugoslavian pressure to align/be subsumed into either nation, at which point in the 20s Albania is formally made a protectorate of the US. then the situation from above continues, with it eventually becoming integral into the USA.
Eh. Even that's farfetched, but I wouldn't say impossible.