You seem and the author seem to maintain it is not plausible to get three rounds in a row in a shot grouping on a single target at 500 yards. I completely agree with this.
Let us say we put a blind fold on and shoot randomly at a piece of paper 20 feet away while shaking the gun, and we shoot many dozens of bullets this way.
If we see a group of holes in the paper closer together than some others, we should not assume they were from a targeted group of consecutive rounds--because obviously we were shooting too poorly for that. But neither should we be surprised it happened. The common sense guess is that some of the rounds out of all ones we shot happened to end up closer than other rounds. Of course the more rounds we fired in the same general direction, the closer these groups are likely to be by luck. The shooter fired a lot of rounds. He may have concentrated on some areas with more bursts than others.
Now the number of hits per target could be reasonably modeled by a Poisson distribution. It is not at all unusual or suspicious with such distributions to have 3 occurrences of one event and 0 of another when the events had equal chances.
Correct, and clustering within the Poisson distribution is well known. It's why, for example, in a locker room three random dripping wet guys all wind up within a few feet of each other with no one else in the room.