"Why the fuss?" "Tempest in a teapot." Etc.

Quite true. And as a Yank I agree. But now you're using reason, which is often a mistake when trying to figure out somebody's reaction to a word.

Language isn't logical. It's psychological.

BTW, it would take a very weird Southerner to be truly (not jocularly) offended simply by being referred to as a "Yank" by a Limey. I mean a Brit. Or an Aussie. I mean an Ozzie.