Think of consequences as being very special cases of fate points. You never get extras, they only refresh based on the level of consequence, and they can only be used in one way (damage reduction). Thinking of them as specialized fate points gives you the right mindset for why they aren't separated into mental/social/physical pools. Consequences are a way to deal with a hit that would have otherwise knocked you out of the conflict, and trade it for a temporary disadvantage to your character. A bullet that would have killed is instead bent by fate, in the form of a consequence, into only scratching the surface of the skin. An argument where all your hopes are about to be dashed simply becomes a case of mild embarrassment, and you can keep arguing on.
The books say that stress is not damage, but a pacing mechanism for how long the character can go without taking real damage. I like to think that consequences aren't damage either. Oh, sure, they do represent real harm done to the character, but its not damage in the Hit Point form of damage or some other analogous thing. Instead, it is damage avoidance.