You do not need accurate time measurements for this method; the question to be answered by each observer is "was the given star occulted on the night in questio

You do not need accurate time measurements for this method; the question to be answered by each observer is "was the given star occulted on the night in question?", and the ancients could track time to the precision of a day (as we are concerned with at most a hemisphere, there's no problems with date lines.) It does depend on accurate surveying over distances exceeding the diameter of the moon, and a good deal of communication, which I suspect is enough of a problem to answer the author's question of why it was not done, as far as we know. The method is not, however, begging the question.