You're getting hung up on the finality of "fatal flaw", and many of these suggestions have a similar problem - the implication that this flaw will inevitably lead to a person's undoing. It's a difficult thing to untie from the concept of one's greatest flaw, since the strongest visual concept of a flaw that great is the fact that it can lead to a person's downfall.
I wouldn't worry about that implication, unless you very strongly want to untie it, because the use of such an idiom in english is usually figurative - a 'fatal flaw' doesn't necessarily mean the flaw will lead to a person's downfall, rather it only implies that the flaw is so great that it could lead to a person's downfall.
If you want to untie the concept of finality from this metaphor, you could try the expression "Key Flaw", which suggests a flaw that is more important and more disarming than any other flaw. And you could also simply use the adjective of 'greatest' to indicate that it is in fact their "greatest flaw". Both of these being a two-word phrase may not make it succinct, but the message of either phrase is quite clear.
As a final suggestion the weakest link refers to the most vulnerable part of a set, so if you're listing a person's characteristics or flaws, naming one as the 'weakest link' would indicate that it's the greatest flaw they have. But you wouldn't individually call it the 'weakest link' unless you have something it would be related to.