I can kind of understand why our hero's first time travel backwards failed to prevent Kayo's murder. He only learned of the exact timing of Kayo's death on the first try after he had changed the timeline so much that it was no longer useful information. However, all he had to do for his second time travel backward was so simple. He knew the exact timing of Kayo's murder down this timeline, from 9:30 pm to 12:30 am on the night of March 3rd. So all he had to do is call the cops and have them stake out the impending murder scene between these hours and they would have caught the killer red-handed. With the culprit caught, the serial killings end, and thus everyone in town is safe.
Even if he somehow couldn't get the cops to believe him and stake out the place, he could have brought his mom along, who could have called the cops and given pictures on the phone recorded evidence of the killer kidnapping Kayo out of the shed in order to kill her. Perhaps this wouldn't be enough to save Kayo, but at least the culprit would be caught and everyone else would be saved. If he had brought the entire gang of kids with him and had them all armed with kitchen knives, even without calling the cops, they could have simply mobbed the suspect before he got to the shed and stabbed him to death. An ambush by five boys with knives would have done in the teacher without any problem, it's not like he's some sort of superhuman.
Instead of any of these reasonable plans that utilize his foreknowledge, he instead tries to alter the timeline again, completely eradicating his own advantage and opportunity, putting everything back to zero again. Now, it's very nice of him to want to improve Kayo's position by getting her out of danger by removing her from the grasp of her parents. But that wasn't the mission. Or at least, that's not the mission right now. Right now his priority is to catch the serial killer so that he can't endlessly go on killing kids for the rest of his life. Kayo is low priority compared to all the victims yet to come. So saving Kayo but not anyone else is a complete waste of time and energy. In his second time trip back, he manages to save Kayo, but there's still tons of other victims ready and waiting for the killer to target. He flounders around trying to save them from being targeted too, but you can't be everywhere guarding everyone at once, so his tactic was bound to fail sooner or later. The fact that he didn't predict himself becoming a target was just one oversight, there would have been millions more, and the serial killer always would have found new prey. It was just a dumb strategy from the very beginning, and it was completely unnecessary because he had the key to capturing the killer right in front of his nose.
Via weird authorial authority, our hero manages to stop the serial killer from killing any more by going into a coma. Why this would stop the killer, who knows, but there ya go. Then when he revives from his coma, he plays weird mind games with the serial killer and somehow manages to capture him without anyone being killed, but again it feels like realistically he should have been killed and his tricks shouldn't have worked. So his method of stopping the serial killer feels sloppy and deus ex machina, but the actual way to stop the killer was easy and foolproof. So from episode 7 onwards the story becomes infinitely aggravating due to this double barreled shotgun of stupidity. 1) He doesn't use the perfect solution that he's been given. 2) The solution he comes to wouldn't have actually worked if the author weren't playing favorites.
Boku Dake ga Inai Machi episodes 1-6 looked like a top 100 series, but due to our hero's stupidity (which is really just the author's stupidity for overlooking such a simple solution) in eps 7-12, it ended up only being 171st. Ultimately, it's because of dumb plot holes like this that time travel should never be used. There are good time travel stories, like Steins;Gate, Higurashi and Haruhi Suzimiya, but most of the time authors don't have a good enough grasp of what time travel entails and how a time traveler can maximize his efficiency, so it always turns out into a squirming mass of contradictions and glaring oversights. It's best if people just don't approach the topic with even a ten foot pole unless they are absolutely certain that they have the subject nailed down to a T.