I've never known anyone, minority or not, who got teary-eyed at the word "slave". Uncomfortableness with the basic terminology of slavery is a reflection of how people relate to their privilege, especially in an era where minority rights and minority power are at the forefront of the contemporary narrative.
That isn't to say that there doesn't exist terminology that people should be mindful of. But most privileged, white people wouldn't recognize it. When the candidate for Florida governor denied using the word "monkey" to illicit traditional racial animus in his conservative base, people on the left were incredulous. But why? As we're seemingly attempting to do with the words "master" and "slave", we work hard to erase the incidents of oppression rather than actually addressing oppression. Is it so hard to believe that, once excised from common speech patterns, the memory of the origins and import of words and phrases is forgotten, even if the undertones linger? Especially among earnest individuals? (You can be earnest but still prejudiced and even prone to using the language of prejudice.)
Now, no doubt blacks in Florida had good reason to be incredulous of his supposed ignorance. But context matters, in particular the perseverance of the more crass language of racism in Florida. The use of the word "monkey" was offensive because of the context and what it implies, quite concretely, about the motives of someone running for a very powerful office that has traditionally done horrible things to certain Florida communities. But nobody is, yet, arguing that we should remove the word monkey from zoo plaques, or stop using terms like monkey-patching.
And that's despite the fact that, very much unlike the word "slave", if you shout out "monkey" in the company of certain groups heads will turn. Chattel slavery doesn't exist and hasn't existed for a long time. Absent a malicious or awkward context, the word "slave" doesn't elicit or trigger any anxiety or anger or tension in African-Americans. That slavery is some pervasive obsession of African-Americans that shapes and colors their every thought is a TV trope, and a racist one at that. Blacks no more identify as a slave then they do a monkey. Again, context matters; it's the context which makes the implication; it's context which turns a word into an instrument of oppression. But when the word and topic legitimately come up it makes privileged whites very uncomfortable, which is why it's so memorable and poignant in their collective imagination, and why it makes for cheap tension in script writing.