Sorry man you are probably the only one in a billion native Chinese speakers that make those connections directly, please don't make it sound like your personal shower thought apply to everyone else.
Normal native Chinese speakers learned immediately that characters used in every single country names are, as defined by "transliterate", purely based on pronunciation, and have absolutely nothing to do with the meaning of the characters.
So native speakers are trained to, and very good at, completely ignore the meaning of separate characters in transliterate words.
I personally have never ever think of American as "beautiful", or France as "lawful" since childhood. Thinking South Korea as "cold" just because "韓" sounds like "寒" is definitely the most hilariously ridiculous connection that I have every heard in my entire life.
I have also never heard of any native Chinese speaker playing these mental gymnastics for country name, or any transliterate word. Never ever read a single sentence that put Mexico and dark skinned people together, or any of those ridiculous examples. Not in real life, not on TV, not in the books, or on the vast Internet.
Note that there IS some strong propaganda going on when government choose the characters of official transliterate country names. Also there are lots of researches around this subject about how these characters affect people's impression of various things.
It's just that in real life, the effect of these country name propaganda on people's impression, is just miniscule and purely subconscious. So in my entire life I have never seen anyone, any material, that speak those impression out loud, so directly, exactly as government wanted, like this shower thought did.
Either OP is a truly unique one in a billion snowflake, or this whole thing is yet another AI generated hallucinating horse shit. That Mexico impression is so, so hilarious that I start to admire the creativity in this post.
Regardless, it's too late now. This misinformation has already reached maybe thousands of people, and they are all curious about what their country's characters mean in Chinese. They will never see my comment, and continue to live with this ridiculous impression of general Chinese speakers, for the rest of their life.