...Unless the lack of Asian people was eventually going to be explained. Maybe there was a genocide before the exodus from Earth.
Why would the genocidal culture seek to incorporate the cultural apparatus of their victims?
(1) They'd already incorporated Asian influences long before the genocide. There may have been peace for centuries before hatred flared, just as the Jews lived in Russia and Europe for centuries before the Pogroms and the Holocaust. At that point, there'd be no way to separate the Western influences from the Asian ones in their mixed culture. Current western cultures are a hodgepodge of influences from disparate peoples who used to be mortal enemies. You probably can't go a whole day without saying, eating or wearing things that were created by the British and other things that were created by the French, for instance.
(2) Westerners didn't dislike Asians, Western GOVERMENTS (that may or may not have been totalitarian by then) decided to eliminate Asians. So there'd be no reason for the average Westerner to want to reject Asian cultural influences.