Factual error: In the newspaper article from the Trinity killers past - the report states that someone called 911 and dispatch gave them instructions. The death occurred in 1959. The first 911 call was made in 1968. (00:39:24)
Character mistake: In this episode, as well as 1-2 others, Dexter holds his slide phone upside down (only in the edits/cuts where he wasn't shown answering the phone). When the phone is slid open, the front cover goes above/higher than the backside of the phone. And a few instances I've seen, the side closest to his face 'lower' than the outside/backside of the slide phone, strongly suggesting the phone was held upside down.
Answer: Two reasons. First, as a cop he knows that there are lots of murderers walking around free due to flaws in the system so he uses Dexter's "urges" to right the wrongs in the system by taking out people who have no right to walk around free. Secondly, by taking him to a psychologist he runs the risk of having him committed and he doesn't want that for obvious reasons, so he figures it is best to channel his urges to what he considers a greater good.
The_Iceman
In addition to the earlier answer, in the last season (8th) of the show, we are introduced to Dr. Vogel, whom Harry had been discussing the situation with Dexter as well, and it is shown on-camera that Harry describes what Dexter has done (killing)... So, technically he did consult a psychologist/psychiatrist, just not having Dexter and her actually meet. But Dr. Vogel was aware of whom it was and how to intercede.