quote OP "he wrote a small bit about how he wished to kill a teacher."
not quite "I'm going to kill this teacher."
It almost certainly listed a particular teacher by name.
Now, some might say the teacher over-reacted, or that if the school takes action they over-reacted. However, the school district and the police in Columbine and additional incidents in the years after got sued because they were accused of failure to take action. It is a "damned if you do, damned if you don't" situation for the school and the authorities.
From my time as an SRO I have to say that binders and noteboooks were outstanding sources of intel, and it takes very little to justify the seizure of such. It does not take a violation of the law to justify a search of a notebook by school staff or the SRO, only a violation of school policy. Look at the binders of some kids ... you will often find sexual commentary, derogatory commentary, gang signs/statements/images on them, etc. It takes very little to justify the search of a binder or notebook by a school official, only reasonable suspicion that a law or policy has been broken.
In this case, there is no evidence to suppress and a crime is not likely to be alleged ... but, the school and the police would be remiss if they did not investigate the threat and take it seriously. If they were to ignore it, and the child later acted on it, they'd be having to write a big check whether they really should be seen as liable or not. Even when you have a good case and can win at trial, forking out hundreds of thousands of dollars is the norm.