The Eastern Rites retained the liturgy in their languages. It was often Greek but also Syrian, Egyptian, Aramaic, etc. These were still used until Vatican II when they too changed to the vernacular. Which may be Greek, Syrian, Egyptian, or Aramaic. Perhaps a more modern use, but sometimes still a form of the original language.
Like translating Spanish into Latin.
Pater Noster becomes Nuestro Padre. An updating of the language into a more modern form. Spanish IS modern Latin, as is Italian (Pater Noster becomes Nostro Padre)
Similarly, Koine Greek would be translated to modern Greek.
What killed the use of Greek (etc) was Islam. With conversion by the sword the eastern church was more than decimated (decimate is from the latin: killing of 1 person in 10). So the west became dominant because the Crusades were able to stop the killing/converting of Christians by the invading Moslems.