To say I've never been afraid or uncomfortable with my weaknesses would be a lie, but I guess I have always felt like I owe it to myself to examine them. I guess it is part of that "striving for truth" thing. I WANT the truth even if it might make me cringe at myself for awhile.
I think the main trick for me has been learning not to beat the living crap out of myself for my weaknesses or lecture myself like I'm some criminal for having them. I think sometimes people just will not go into that territory because of how much criticism is just floating around out there toward weaknesses, whatever they might be.
I grew up in a very critical family myself, so I think if one has grown up feeling like they had to apologize for so much it can be a challenge if someone asks them to examine weakness willingly. Not to mention that for every weakness you face up to and deal with, there are seemingly three more to come after that sometimes, hehe.
Really though, I think sometimes that a person's unwillingness to face weakness might be that ingrained things in a lot of families that 'weakness = failure'. It's destructive but it's out there, you know?