Hold your hand on front of your face. Now call it the worst names you can image. What harm befell your hand? Nothing. Words have no power unless you give power to them.
Not sure that is the best analogy. In this case, if I give power to those harmful words I spewed at my hand, then harm will fall upon it?
Maybe the simple “sticks and stones” mantra is the easiest.
I like your tagline. Groucho was always my favorite.
It's not the words themselves, but the ideas described or associated with those words. For example, a number of people will be offended if you use "crude words" to describe sexual activities, sexual elements, or body parts associated with sexual activities. Black people take severe exception to non-black people (not just "whites") using the slang word for "Negros". (And even that word will cause Blacks to react strongly nowadays.)
Firesign Theater had a song that put a number of so-called "hateful" words into a verse, with the end of the song being, "That's America, buddy!" George Carlin had a monologue, "seven words you can't use on radio or TV."
"Sticks and stone will break my bones, but words will never hurt me" -- that's very true.
Stand up in front of a crowd and start ranting that a certain group of people is responsible for all the evils in the world. Do this repeatedly.
What harm would befall the group of people so targeted?