Around the time of WWI, there were probably more Rabbonim from Poland, Lita, Ukraine, or Russia, than from all of Western Europe combined. Germany just had it’s best period for the Jews. Eastern European Jews had been steadily migrating for forty years.
There were over a dozen Chassidishe Rebbes that came to America before Rav Moshe. And a similar number of yeshivos. The problem was the assimilation of the second generation immigrants. It had nothing to do with the brand of Orthodoxy.