A capacitor can't 'dump' it's energy any faster than tbe circuit attached to it can use it, as illustrated by Ohm's law (ignoring the capacitor's self discharge rate, which for a purpose like this can be kept very low).

With a low self discharge rate there's also nothing brief about the length of time a capacitor can store energy. Think of it like a bucket with a small leak (self discharge rate). We're adding water/energy to it each day (from solar panels or similar), and taking some out as needed. As long as the leak is small in comparison to the amount of water/energy being put in it'll still be well topped up when you need it.

With regard to a crack, it sounds like that will just lead to a reduction in storage capacity to worst, if a chunk actually falls off. The carbon forms a branching network throughout the entire block, so just breaking some strands wont stop it being conductive. You'd definitely want to deplete it before working on it though, even a small cap can give you a nasty whack.