You certainly may. In fact, we'd hope you do.
I agree with you that it was fun to learn the language a little and refer to them in their native tongue back in the Ancient era of Rome Total war and it's mods. But I feel like now it's a bit more complicated. In RTR or EB you had the somewhat 'universal' greek, being found in many factions (In RTR, in almost all of those of the civilized world excluding the armenians and persians. In EB, not so much but still very signifigant), so that toxotes = bow or kontos = lance became reinforced.
But in the medieval era, while greek and latin were international languages spoken by all sorts of people of faith and culture (Although greek is probably the dominant thing in the east, and latin in the west), it's not what dominates the military lexicon. The closest would be Arabic or Turkish, but as far as I know it was usually not all or nothing but rather a blending of the two. I know it deals with them, so I'll look it up, but in general, it'd be more difficult and more complicated to feature wholly native tongue units in Broken Crescent than it would be in an Ancient Era Mod.
Someone coming in would have to end up recognizing Greek, Armenian, Georgian, Arab, Turkish, Mongolian, and French terms. While our Georgian and Armenian contributors have given us all the information we need on naming their units, and we can find someone to do the same for Byzantium, it'd be harder to do for the other factions.
All in all, it would be nice, and I do hope that MTR and MA feature it. As for us, we are somewhat of a Realism-based, but not restricted mod, so unless the mighty Caliph Mirage (Blessings of God upon him...except on Friday the 25th. ) should happen to have a change of heart, or should happen to fall from the parapets of the Citadel...we'll probably stick with half ethnic and half anglicized.
Perhaps what we can do is when it is applicable (For Byzantium, Armenia, Georgia, and if we have the information, the rest), is feature the ethnic name in the description, reversing what you suggest about having the anglicized name in the description. This way, we can educate without impeding.
And what's always possible is for, with permission, one of our members or one of our fans to take the name file and replace the half-and-half names with fully ethnic names for those factions which we have the info for.