“The short end of the stick” makes no sense. A stick could be “long” or “short,” but how can the end of a stick be one or the other? And if an end did have a length, wouldn’t it be the length of the stick, making both ends the same length—so that you couldn’t tell them apart based on length?
More importantly, how did I go this long using the phrase without pausing to wonder at its incoherence?