Aside MC testimony there was nothing saying that Himekawa got to school that day, if they didn't find about it on the same day it happened the culprit could have probably hidden all traces.
In Japan if the police isn't 110% sure about who made the act is possible for the criminal to walk free, they want to artificially keep the number of criminal incarcerated without the system doing a mistake at 99% (or around that), cases where they don't have enough can simply be dropped and the bastard would walk free, so it make sense to rush on the first chapter, it even had to give a general introduction and make us understand the style of this author.
There is a reason that MC testimony about it could have not be enough, you will understand in coming chapters, the relationship Himekawa had was hidden too and there were probably no physical proof, the rumors about his sexuality could have helped push a rumor of him running away or someone killing him out of hate, the best ending was a clean one in the same day it happened.