Well, I personally think that as soon as Morris family gets the whip there is from a storytelling perspective, not necessarily from a pure logical perspective, there's no convincing reason why they would stay in the USA. Of course you could say they are reluctant to meet their destiny and so on, which makes perfect sense in itself, but when you think about the story mechanically, it would just be kind of pointless. You would basically stall their entrance on stage just so that OoE can happen.
It would also like to further defend my stance with a piece of lore introduced by HoD, which is that whenever Dracula revives, all his lackies that got killed off come back as well, and that the Belmont family has the additional duty of hunting them down whenever Dracula is not around. So after SotN we still have a bunch of monsters running around in Transylvania that need to be taken care of. So I think it would be weird if the Morris family would have accepted the whip they wouldn't immediately go after these monsters otherwise they would not fulfill their duty as successors to the Belmont family.
Of course the duty of destroying these monsters ended up in the various organizations like Ecclesia. So I personally think that when these organisations were around, the Morris family wasn't, not even somewhere else out of sight, because it raises unnecessary questions and doesn't make sense, IMO. Whether that's actually the case is a whole other question of course.