I don't know. It doesn't matter. Plenty of saints have sung in these modes, that's for sure.
What do you mean you don't know, it doesn't matter? That's silly and rude.
Byzantine is the only music of The Church and it has eight modes. It sounds a bit different according to where it is sung, Romania, Greece, but polyphony and folk elements are to be avoided as they are secular in origin, not divinely inspired. As far as authentic Byz music has ever got is using two voices in a very particular way.
How about the famous hymn Agni Parthene? It isn't "Byzantine" (it is Byzantine in style & inspiration, but isn't proper Byzantine as typically done) as you say, yet it was composed by a saint (St. Nektarios of Aegina).
You are so wrong in your assumption. There isn't an official chant of the church, just as there isn't a "holy language" of the church. If you learn the history of your church you'd learn this.
Byzantine co-existed and currently co-exists with other chants like Old Roman Chant, Gothic Chant, Gaelic Chant, Ambrosian Chant, Gallican Chant, Cistercian Chant, Mozarabic Chant, Coptic Chant, Gregorian Chant, Russian Chant, Romanian Chant, Serbian Chant, Georgian Chant and Armenian Chant.
Byzantine was just the style used in Hagia Sophia and eventually spread to other parts of the empire. However it was never "the" chant, nor was it the official style. In fact, the style of Byzantine we have today is a much evolved and changed form of the original.
We have hundreds of hymns and melodies composed by people we don't know the names of. St. Romanos and others are just a few we know the names of, and they didn't even compose the majority of our hymns.
So, why do they use and develop only Byzantine music on Mount Athos, regardless of ethnicity? What we are saying here is that there aren't any rules, anything goes really.
Mount Athos is in Greece. Does Mount Athos represent all of Orthodoxy, or set the rule for all Orthodox Churches? I think not. Mount Athos uses Byzantine Time, does that mean we should also, or that Byzantine Time is more holy than regular time?
You are falling into the "slippery slope" fallacy here. Just because we are saying Byzantine isn't the only "holy chant" doesn't automatically mean we can start using instruments and "contemporary" worship in our services.
In Greece, liturgy is in ancient Greek. Does that mean we should all have our services in ancient Greek? Or that ancient Greek is more holy than modern Greek? Certainly not! We aren't Roman Catholics, we don't believe in sacred languages. Wherever Orthodoxy is taken, there the services are always translated into the vernacular language (which ancient Greek originally was). In the same realm, wherever Orthodoxy is taken, the people there eventually form their own style of chanting, which may or may not be similar to Byzantine Chant. Again, in the same realm, wherever Orthodoxy is taken, the people eventually form their own unique & traditional architectural style. It may or may not reflect 14th & 15th Century Byzantine architecture, but it is nonetheless holy.
Our Church isn't the Byzantine Empire frozen in the 1400s. It is a living and growing tradition.
As I'm in the profession of architecture, I can draw an analogy here. Traditional and vernacular architecture (i would argue) is the best and most beautiful form of architecture. However, we aren't frozen or bound to the classical tradition of ancient Greece, or ancient Rome. We have the ability and the duty to expand the classical & traditional language. However, that doesn't mean we have permission to do as the modernist architects did 100 years ago and completely throw out tradition and start an architectural free-for-all. Whether its Classical Greek, Classical Roman, Romanesque, Byzantine, Neo-Byzantine, Neo-Classical, Baroque, or Vernacular it's all beautiful and the language can and has expanded without being completely thrown open.