"Even if the phaser shot was the result of impatience, it would not have caused the death. Please look for blood or discolouration of the skin, particularly around the head and neck, or the abdomen. On humans, and others of red blood, the discolouration should likely be a shade of purple." And why would a poisoning victim end up in a firefight with their murderer, a fight that would injure but did not kill, and then still die in such a highly public place? Logic dictates a phaser burn would require medical attention, so a visit to the med bay would be required. Any poison that was fast-acting enough to kill in such a short time would also not warrant getting impatient over, unless the killer was absolutely useless.
This was a game. He wasn't sure he could discount the possibility of the killer being useless. But it was still a possibility he didn't want to entertain quite yet, until other clues pointed that way. The far more likely option, at least so far, was that Shoulders was the useless one.