Bullying has been known to be a severe, yet unspoken chronic issue in Japan's schools. The bullying you see in other manga may seem exaggerated and unrealistic, but reality has a way to make things stranger. There are a number of reasons why this topic doesn't get breached upon and hasn't been attempted to stop, but I think one key factor here is reputation.
Japan's society is based on a hierarchy. It's seen in how people addressing themselves with specific suffixes or titles, solidifying where the two participating parties stand, be they senior-junior, teacher-student, parent-child, and even boy-girl. Those who are said to have more experience have a reputation ingrained in them to uphold, to set an example to those who are seeking some form of guidance or goal to reach. This value of reputation can also be reflected in the type of administrations competing against each other, such as schools, not just for academic excellence, but for student life, successes in extracurricular activities, and of course, the number of reported cases in delinquency, bullying, and suicide, to name a few.
Of course, nobody would want to go to a school where such oppression is rampant. This would lead to less students attending, meaning less funds to put into the student curriculum, making school even less fun, and giving more reason for prospecting students to not attend, thus cycling more and more as the stress builds up until the school has to be forced to shut down. So how do they combat this? How do they make the school presentable enough for students to attend and make it a success? They keep quiet, pretend those things don't exist, or at worst, see them as a form of playful roughhousing. They won't present the problem until it comes up in a way that it couldn't be ignored, or their school's reputation will be put on the line. Even if someone reported a complaint, no matter how vague the message is nor the legitimacy of its source, the school would have to act on it, to show that they 'care' the students enough to resolve the issue themselves as swiftly as possible. They may be lucky to find the source, if they really took their time investigating, but they'll probably put in the minimal effort to show the public eye they're doing something about it, until there's no further traces of the problem and give up, saying they've tried.
Take this with a grain of salt as this is the result of me consuming many Japanese media over the years showcasing similar themes and learning what I could of its culture when it's convenient/legitimate enough. I'm just trying to make sense of what the bitch ex is aiming for and why her plan may fall through to a plop if nobody does anything about it, which is highly unlikely here, or there wouldn't be a story. I'll be curious to see how the teachers here tackle this anonymous letter reporting a bully sitting in that very classroom and try to danganronpa its way of finding the 'blackened', when in reality the culprit isn't even in that room.