Confusion about gravitational potential

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I thought I know introductory physics well but now I'm confused about simple things!
Imagine a pendulum. Its released from rest from a height. We take that height to be the zero level of the gravitational potential. But that means its total energy is zero at the beginning and because of the conservation of energy its going to remain zero and so there is no motion. But if we choose another height as the origin, then the energy is non-zero and there is motion. But what happens shouldn't depend on our choices. What am I missing here?
 
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Recall that it is only changes-in-potential energy that are physically relevant (not the values themselves).
Today, I made the analogy that the position of your head (as read by a vertical meterstick) is not a physically meaningful measurement of your height. Instead, it is the change-in-position from your feet to your head.

"Zero Mechanical Energy" doesn't mean "no motion" (K=0 and delta-K=0)... it means that K+U=0.
If (delta-E)=0, then (delta-K)= - (delta-U).