As I am from the Korinek family and also from the Czech Republic, I have my own idea about this topic :-)
I don't think that all Korinek's in USA are related. Here in Bohemia and Moravia was quite a lot of Korinek families as early as 17th century and we don't have any relation between them. So if some families were emigrated to the world, we cannot suppose they all are related - if we don't have documented they are from the same region.
And because a lot of emigrants went off homeland with wish to start "new life", I suppose they didn't keep many information. So in passenger's lists are mentioned as only from Bohemia (or Moravia or Austria etc.). And in that case it is hard to find the right region, which is home of relevant family. Maybe via DNA tests? :-)