How can the centripetal force lead to objects flying apart?
In the lab frame of reference, you need to reverse the question - don't ask yourself what pulls the particles apart but what keeps them together.
By Newton's laws, everything on which no force acts keeps travelling in a straight line. So what requires explanation is not that a collection of moving particles - such as a rotating flywheel - flies apart but what keeps them together. The force that keeps them together is a centripetal force, in this case exerted by the bonds that keep the material together. When you reach a velocity where this force is not enough anymore to keep the particles on a circular trajectory/bound orbit, they fly apart.
Try to imagine, instead of a big stone, a big plate. On top, you fill the plate with sand. Now you start spinning the plate.
What's going to happen to the sand? The sand is going to leave the plate very fast, and spill in all directions. This is the basic, natural state of things, and it's from here that you should start questioning.
How do we keep the sand from leaving the plate? The answer is the centripetal force. If you tie each sand grain to the center of the plate with strings, when rotating each string will pull on its grain of sand, and prevent it from leaving the plate.
Using a big rock is the same thing; instead of using strings you're just relying on the intrinsic cohesivity of the rock with itself. Here each "grain of sand", or piece of the big rock, is attached to the pieces adjacent to it. They in turn transfer all forces to the ones adjacent to themselves, and this is how the centripetal force is propagated from the outside to the center of the big rock.
Once any piece of the rock fails to sustain this force, the bonds break. Once again you have sand, and as we said above, sand is going to shoot in all directions.
You don't have to explain this by centrifugal force, or any fictitious force at all. All what centrifugal force is about is inertia.
As your stone is spinning, it has some velocity. But since initially there's a centripetal force, this velocity constantly changes towards the center of rotation. When part of the stone breaks off, it's no longer held by centripetal force, so it just flies due to inertia with constant velocity.
It's just when you go to the frame of reference associated with the stone, only there you get centrifugal force — as a device to make Newton's laws look unchanged.