The song uses the epigram in a bad, wrong, way. First: it is a epigram, a "a brief, interesting, memorable, and sometimes surprising or satirical statement", not any sort of philosophical argument. Nietzsche liked the format to get down simple statements that were meant to get the reader to think, not to lead to any definite conclusion. (The full epigram is ""From life's school of war: what does not kill me makes me stronger". "What doesn't kill you makes you stronger" seems to be a 1970s paraphrasing by G. Gordon Liddy, of all fuckin' people.) Other epigrams from "The Twilight of the Idols" include:

"Can an ass be tragic? To perish under a burden one can neither bear nor throw off? The case of the philosopher."

"What? You search? You would multiply yourself by ten, by a hundred? You seek followers? Seek zeros!"

"I mistrust all systematizers and avoid them. The will to a system is a lack of integrity."

(Note: I'm, uh, skipping out on some of the sexist ones also there. Nietzsche was considerably ahead of his time when it came to philosophy, but behind 1880s Germany when it came to women.)

Point: Nietzsche never meant it as a real philosophical argument.

Second: if you went around telling other people "What doesn't kill you makes you stronger" you'd be an idiot in Nietzsche's eyes, because you would be using the statement as a moral injunction. TL;DR one of Nietzsche's big philosophical projects was disproving the existence of certain types of morality, one of which can be summarized "one course of action that is always good for everyone, all the time." (In the Twilight of the Idols, Nietzsche likens this to dieting, and how there is no such thing as one specific diet that is right for literally everyone. This is because we are all different, and furthermore, do different things in our lives, and it is the personal context that matters, not some formula that worked very well for somebody. He could write for 1880s Jezabel if he wasn't churning out epigrams like "Women are considered profound. Why? Because we never fathom their depths. But women aren't even shallow.")

Point: For those of you that have heard this epigram used this way and thought it was profoundly dumb, well, Nietzsche agrees with you.

(PS: I laid all this out for a friend once, and her response was "Gee, I wonder why Kelly Clarkson didn't consult you?" I have jerk friends.)