What all runners start at a common origin, their phase are the same. When the phase of the system is not considered then the runners simply pepper themselves on the path at various time and start at various phase.
This is the situation that the proof is for, where a snapshot was taken, that given a initial least distance the next least distance requirement is also satisfied.
What else is a common origin for? Does it matter if one guy is late. Does the system forgets. Does the reference for which this analogue is based show concern for phase and starting point?
