I guess I am ambivalent about the place of the Western-rite. I think the liturgical nuances of the western-rite, especially the Sarum revival being attempted by some has little socio-cultural connection to Western people. The approach of the Antiochian Western-rite Vicariate in drawing on post-Reformation Latin Catholic tradition - Tridentine as lived by the Western Latin Church into ther 1960's offers a much closer cultural connection.
If we look at Australia, there has been a complete failure to connect culturally and spiritually with seekers, which is why the Western-rite has no parishes or missions in any of the major Australian mainland cities. Whether Western-rite or Byzantine-rite, the Church's evangelism is increasingly to those who have minimal connection with the heterodox Christian history and culture of their country. The success of the gay marriage movement among the unchurched mainstream has occurred on the back of an acceptance of non-marital conjugal life - de facto relationships where people simply choose not to get married. That if course is closely tied to widespread acceptance of sex life outside of heterosexual marriage.
Relevance to people's lives is what counts in evangelism. I think this is why the 'Eastern' Orthodox have succeeded where Western-rite missions have been at best patchy. People connect with people - and liturgical language, or whether a rite is Eastern in origin is pretty irrelevant. A good pastor who can touch hearts and lives will have an impact in his community that will enable people to consider the Gospel and the Orthodox Church - whatever the rite.