On a simple level of interpretation, there was another cause for waiting until the entire eastern sky was lit, until Hebron. It once transpired that the priests in the Holy Temple actually did make a mistake in determining the day's commencement - and the daily sacrifice was offered too early, unintentionally violating the Biblical requirement.
The Mishna in Yoma 2:2 relates that upon that occasion, it was a cloudy day, towards the end of the month (in the Hebrew calendar, lunar months are observed. At the end of the month, the moon rises in the sky close to dawn). The moon's light, which broke through the clouds and lit up patches of the sky, gave the impression that the dawn was at hand and that the eastern sky was aglow with the sunrise. The daily tamid offering was slaughtered, and afterwards the priests realized that this had been done while it was yet night - and was therefore invalid.