Is the INTJ drive for mastering skills really about gaining approval, or is there something else behind it?

How do I know if my gut feeling is true or not? I've been having a thought for a long term and I just can't get off of it. Is it really this true?
Do INTJs really get called arrogant or show-offs just for being skilled, and how do they deal with that perception?
How do INTJs handle friendships when they often feel betrayed if their friends make new connections?