None of those notes about Seo really sounded bad enough to kill someone over? The two that were covered might be, but you can't really read them. I would have put "Seo killed my cat and laughed" front and center if I was trying to convince people Seo was a bad enough bully to deserve to die.
The way the letters are worded seems that people just assumed she did those things or at least that's how it is in the unofficial English translation. Seo was really mean to Satsuki but who knows if the accusations were true or if they were exaggerated.