pins are usually the first component in chains to fail by sheering for the most part. (much like climbing caribiners, they'll fail at the pin for the gate hinge when overloaded)

the other big weak point is a chain connected via a rivet. often alignment is off just enough to stress the side plate when resetting the pin.

all in all, chains don't fail often. most efficient way to transfer power. iirc they're about 95% efficient in transferring power.