I agree with the variety comment but why can't there be both in the same training setting? It is true that it would be a taller order for the instructor(s). I certainly feel using a styles native language makes sense if a school/instructor uses the "traditional" moniker.
As to your question, there does seem to be less usage of Korean terminology in the TKD circles I run in. Ironically, in sport TKD all commands and judges calls are made in Korean.
Occam's Razor might suggest that Korean is a lot harder to pronounce than Japanese if you don't actually know either language