I strongly disagree with this statement, the music is out there if you were to look.
Jazz has moved past the standard 'swing and ballad' forms that were popular throughout the late 20th century, and most listeners can't shake the roots of the genre from what defines the modern equivalent; It would be akin to die-hard old school rap fans denying that rap in it's current state should even be considered rap.
>That is what makes singing and rapping unique, you are able to tell a story combined with music.
What makes jazz (particularly the instrumental kind) unique is being able to invoke emotions and tell a story without being bound to some language barrier, an abstraction that transcends spoken word, that speaks to the pattern matching automata of the language part of our brains, regardless of race, language, or background.