I would replace "During each upkeep phase, instead of drawing 1 card, draw 2 cards and discard 1 of them." to:
"During each upkeep phase, instead of drawing 1 card, look at the top 2 cards of your deck. Draw 1 and any weakness among them. Discard the rest."
It seems fair middle road to me. I think both other interpretations are unbalanced. If you draw a weakness with it, then allowing to discard that weaknesses (right after the revelation) is too powerful if it's a crippling asset or enemy. Then there's the other way, forcing a discard of the other card makes it near unplayable. You would not only be getting the activation of a weakness, but also losing a (good) card!