I think you have a good point, but FWIW, you almost never see 中国 in Japan being used to refer to the Chuugoku Region - I’m racking my brains and can’t think of a single time I’ve seen it being used without 地方 attached in that case (and my partner is from Yamaguchi so I see / hear the district mentioned more than most, I guess). Most other regions are used standalone - 東北, 中部, 近畿, 北陸 etc. - but Chuugoku always gets 地方 attached (this is also common with 近畿, but not always the case).
I can see the argument for not teaching these two conflicting definitions, since it’s a pretty early piece of vocabulary and it’s not generally used in that way in Japan… But I definitely agree that if 地方 isn’t in the vocab list, it should be (it’s used a lot in everyday Japanese, especially if you’re travelling etc.), and 中国地方 ought to be a vocab term in its own right.