"Only the distance from the pivot to the center of gravity is relevant"
I'm quite sure this is incorrect. How 'fast' the rod falls is determined by its angular acceleration. This is determined not only by the distance from the pivot to the center of gravity, but also by the rod's moment of inertia.
If a uniform rod of length L and mass m is tilted at an angle A to the vertical, then its angular acceleration is 3g(sinA)/2L
This shows that a longer rod will fall slower, and so is easier to balance.
Now imagine putting a steel ball of mass M at each end of this rod. Note that its center of gravity does not change, but it's moment of inertia does. It's new angular acceleration is (3g(sinA)/2L) x (m+2M)/(m+3M).
This is smaller than the old angular acceleration, showing that placing a mass at each end causes the rod to fall slower. The bigger the mass, the easier it is to balance.
by James Hennigan, in response to this post 2016-03-30 19:13:05