There is nothing inconsistent with being an introvert and joining a club. I was in the chess club in high school – we didn’t compete, it was just a place I could go to find other kids who were interested in playing chess and maybe improving their game somewhat. That was in the days before computers, so if a kid wanted to play chess they needed a partner.
I do think the word “introvert” is being used by many to describe a different set of qualities, including avoidance and reticence. Introverts are not necessarily loners – they may be, but more common that introverts will have a small set of close friends.
Introversion has nothing whatsoever to do with the ability to take initiative,become engage, or pursue interests, including arts and athletics. For example, high school orchestras are full of introverted kids who are quite willing to participate in the organized group – though each student sits alone with their own instrument throughout practices and performances, and musicianship generally involves hours and hours of solo practice.