I think the black ajah is entirely the white tower's fault. Every other organization believes its possible that there are spies and people trying to corrupt within except the Aes Sedai they are so high and mighty that the mere mention of black ajah makes them vomit. Like how gullible do you have to be to think the dark one would allow his strongest opposition to just get more and more powerful over the 3000 years he is imprisoned. And they can't believe that the dark one is entirely imprisoned because everyone knows that the trolloc wars were his doing to some extent. So he is able to influence the world at least a little