Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C* y.
It follows that z C y but it does not follow that x C z or that y C z.. The "whole" z may be a cause of its parts, without in turn being caused by its parts. Note that by construction of C it is true that if x is a cause of y and x is a part of z, then z C y. However, it is not generally true that if x is a cause of y and z is a part of x then z C y.
As an example of the intuition behind this: suppose I have a thermostat box containing two circuit boards. Board 1 is connected into my home heating system; Board 2 is a spare not connected into anything. It is true that Board 1 causes my heating to come on. It is true that the thermostat (of which Board 1 is part) causes my heating to come on. It is false that Board 2 (which is part of the thermostat) causes my heating to come on.
But then we have a longer chain; using x C z C y in place of x C* y.
You are right that when adding z, we now get a longer chain {x, y, z}, but this won't in general be an "endless chain" (the new z may well be an end).
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C* y.
It follows that z C y but it does not follow that x C z or that y C* z.
It does, because y P z.
x C y, and y P z. Therefore, x C z.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.