the full story is not anything like, "other people have worse problems so STFU"

I think the problem that people are having with the quote is that it doesn't actually contain the full story, and when it is repeated outside that context, the meaning they get from parsing the words is "other people have worse problems so STFU", and it's not a good idea to go around repeating it if people are going to predictably lack the context and misinterpret it.