Disclosure: I'd probably test as an extravert, but would advise against identifying as an introvert in a business setting due to the bias against introverts.
I don't think any undue stereotypes around the introversion / extraversion dimension come from Jung, but have simply been attached to the concept by ill-informed society. It's hard to fault Jung for this. Introversion / Extraversion is also part of the Big 5 and has more substantial scientific backing than the foundations laid by Jung.
There's lots of things about me, that I can reveal to some people and not others. I want to be in control of the narrative.
Astrology is much more obviously incoherent, but it isn't going to "collapse under its own weight" any time soon either.