While I know that most seasonal animal activity is phototrophic response (caused by length of day) like shedding, molting, nesting etc., I also know that the moon's gravitational pull is know to have an effect on some behavior. After all, the moon is responsible for tides which move oceans enough to cause changes in sea level measured in feet.
Having been a subscriber of "Deer and Deer Hunting" for years, I am familiar with Charles Alsheimer's work, and have seen him speak in person. He is undoubtedly an expert in the behavior of Whitetail Deer. I have read his peak rut projections for several of the 16 years he has been making them, and found them to generally be true. I read his first article detailing the "rutting moon" and have followed it with my own experience, noticing more activity around the forecast times both in person and on trail cameras.
Whenever a new theory is proposed, there will be comments from both sides, and often the most vocal in opposition to it. I don't think Mr. Alsheimer is suggesting that one limit their hunting time to only the projected "peak", but rather that activity has the potential to be greatest during this period. Everyone is looking for that one magical trick that will remove all chance and make their hunt a sure thing, when the only sure thing is that one doesn't exist. The only real way to increase your odds of success is to spend more time in the stand.