The Byzantine Rite is ancient and beautiful. But so is the Latin Rite. Moreover, it evolved specifically to speak to the Western soul. In the same way that you wouldn't force Russians to use Greek chant (I hope), now that they've evolved their own, there's no need for a sort of Byzantine chauvinism that tries to deny the West its own patrimony.
I find the idea of a 'Western soul' pretty questionable. Mediterraneans, Germans, the French, Brits, Scandinavians are very different, not to mention Americans of various stripes, Australians, Canadians, etc. For example, the idea that a Scandinavian should instinctively respond more favourably to the liturgical tradition of a Spaniard or Italian on the other edge of Europe (who are culturally far more similar to Greeks than to northern Europeans), than to the liturgical tradition of the Russians just across the border, just because of some notion of being "Western" is nonsense.
Yes, the Latin rite will seem more familiar to those Westerners who grew up with it, or rites derived from it, but in many places, certainly in northern Europe, any kind of liturgical rite is utterly alien, and their "Western souls" will be no more receptive to a Latin rite than a Byzantine one.
Znamenny chant is uniquely Russian, and has a much longer history of continuous use in Russia than the modern four-part harmony does. Yet if you would ask an average Russian what s/he's think of returning to Znamenny, most (at least in my experience) would be dead against it. But based on the pro-WR arguments for resurrecting rites used 1000 years ago, the "Russian soul" should prefer Znamenny over Germano-Italian choir music.
Good or bad, the notion that Westerners should follow the Latin liturgical rite is an ideological one.