I'm going to agree with the previous poster--the Unspeakable Trio are certainly top contenders for worst weaknesses in the game, particularly Bloodthirsty and Unspeakable Oath (Cowardice). If you happen to draw one into a deck that is set up for it, great, but obviously I'm not talking about that.
What makes them so bad? Perhaps we're supposed to say they are boring, because it's considered bad form to complain about difficulty ("Don't play arkham if you don't want a challenge!" There, I've said it for you, so now you don't need to comment.). However, I don't think they are boring. I think they are frustrating and demoralizing in a way that I consider bad game design. They can be extremely resource intensive to manage, requiring expending resources and time to draw and play special cards to handle the thing and to take the actions required, which are tests you have to actually succeed at to finally discard the dang thing. So, they are disproportionatly difficult in terms of investment compared to weaknesses that require just spending two actions and being done. But there's a lot of unevenness in the weaknesses-- Paranoia and Amnesia are also pretty brutal.
These get moved to the "bad design" category for me because they are directly destructive to the main mechanism of player engagement with the game: deck building.
As the poster above noted, as an unlucky draw, you have to redesign your deck around these. Chances are good you've invested a lot of time into designing your deck, because that's your main opportunity for choice, control, and experimentation. It's your place to "shine." It's also the place you get to decide what kind of character you play. When a player has chosen to play an all-fight tank with agility 1 or 2 and they suddenly have to be able to evade a monster twice in one turn, you've taken away their choice about what kind of game experience and fantasy they get to have. The player is forced to clog their deck with stuff to deal with the weakness, perhaps significantly compromising their build.
If you were the dungeon master for a game where your player was super excited to play a wizard, and you consistently targeted them with challenges of physical strength, you would be a bad DM. That's what this is. It's major side quest in every single scenario for a character type you didn't want to play.
And what's the consequence of failure? You hit them in the deck AGAIN, preventing the player from enjoying the rewards of a hard fight by depriving them of XP to upgrade their deck. Now you've made their game harder, their character lamer, and they are falling behind the rest of the team in capability. Removing the reward for a TEAM success and generating a feelings of envy and disconnect, is simply bad design in a cooperative game, especially since no one else can help you with these cards.
"So just house rule them." Thanks, I will. I was just annoyed enough by the card design that I wanted to actually post an analysis of why I think they are bad cards.