Specifically, I'm talking about Music Bands, Choirs & Orchestras would play. As a Choir Student, I've sung my large scale of European and African pieces, but I've never sung nor heard a song from Asia nor from Native Americans. Are there any Historical explanations for this, or am I not exposed to enough variety?
Music Bands, Choirs & Orchestras
That sounds very very European... Most of the standard repertoire was composed by European musicians, it is still very normal to find a lot of their music if you are into the Western musical tradition (even if you are somewhere else in the world).
European concert music certainly made an influence in lots of places, and we find composers from the times of the European colonies, plus music composed once countries became independent. These days we can find new music in the classical tradition being composed by musicians from many different cultures.
So called classical music has had a big "museum" component in the last 150 years. Mozart is a household name while many other composers are not known even in their native countries...
If you are after old music in the Western tradition composed by people born out of Europe, I can tell you about some "Mexican" composers:
There were quite a few musicians working in what is now Mexico who were born in Spain (and maybe somewhere else in Europe), their music is now considered part of the musical practice of Mexico from that period (some lived most of their lives here). For example:
Hernando Franco. Born in Spain in 1532. The text from that video is in Nahuatl, I think.
Juan Gutiérrez de Padilla Born in Spain in 1590.
Ignacio Jerusalem y Stella Born in Italy c. 1707.
We suspect some composers from the period were not of Spanish descent, or at least were not 100% Spaniards.
Getting closer to our days
Carlos Chávez (b. 1899) A piano concerto.
Silvestre Revueltas (b. 1899) Orchestral music.
José Pablo Moncayo (b. 1912) this is probably the most famous orchestral work by a Mexican composer.
Arturo Márquez (b. 1950). This is his most popular work, composed in the early 90's, a live performance can be quite an impressive thing with all the brass and percussion involved. It has this visceral punch...
Márquez is still composing. I heard the premiere of a choral-orchestral work a few years ago, here he used famous speeches. There's this VERY strong movement with the text being some parts of Dr. King's I have a Dream. In my opinion, the best part of that choral work is the one with a baritone, based on a speech by "an Indian chief." I am afraid I cannot find recordings of these...
There are lots of Mexican composers with quite dissonant compositional languages, too. I just went to list easy to listen examples.
I mentioned Mexican composers because, being Mexican, I am more familiar with those. But there are examples from other parts of the world.
Chen Gang, He Zhan-hao Violin concerto composed in 1959, in China.
Alberto Ginastera (b. 1916) A piano sonata by an Argentinian composer.
TL;DR
It was "dead white men" for a while, but there is certainly more to it. The Western musical tradition has been absorbed all around the world, you can find examples from many places. Ask your teachers and conductors, they might be able to give you some suggestions to start listening to less famous people. If they aren't into that kind of thing, I am sure it will be easy to ask in the musical subreddits for more suggestions.