All that which you called filler was meant for saving Hinazuki. His hypothesis was that Hinazuki was targeted because she was a bullied loner and he would simply need to change that and get her past the exact original date of disappearance. So, he made her a normal kid in that sense, with friends and things to do, instead of just an empty park where she would loiter to avoid going home. Sure, he still did things wrong which is why he failed. But in the first place he was no super smart hero. He was a failure at life, more or less. He only solved the previous time jump cases because they were incredibly clear, like that one case demonstrated where he jumped back some tens of seconds at max and immediately spotted the sleeping driver. No need to be Derrick to solve that case. I'd assume the other cases have been more or less similar.

People do stupid shit all the time because they believe they know what they are doing. Many of the worst accidents caused by humans happen due to that.

If I don't recall entirely wrong, the teacher said officials have visited the Hinazuki household a couple of times already, but the mom is cunning and came out clean.