From my storehouse of unsubstantiated factoids, it occurs to me that the huge majority of Yiddish speakers in the United States at the time were not from Germany but from Poland, Russia, and other Eastern European countries. Jewish friends of mine (now over 60), who are first and second generation Americans with ancestry from those two countries specifically, had parents and grandparents who spoke Yiddish in the old country. They have intimated, in a kind of humorous fashion on occasion, that so-n-so is a German Jew, you know ... they come from a better background.
Sobiest’s color-blindness analogy suggests to me that as cultures crossed the Atlantic they were altered in ways we can’t always calculate. Some aspects were increased, some decreased. Did German Jews feel more allegiance to German Catholics and Protestants, or to Polish and Russian Jews? Would a Yiddish speaking Pole and a Yiddish speaking Russian team up with a German Jew who only spoke Hochdeutsch? We’ll probably never be able to analyze it with statistical certainty. It was quite a mix, and individuals will often do what they want without reference to the trends of history.
I suspect, however counter-intuitive to the above statement about history’s incalculablity, that the Jewish influence on the word gesundheit is separable from the German influence because Yiddish and German represented more than sufficiently distinct cultures outside the question of religion.