
They’re making her “feel bad” because of what she is: nosy. It’s none of her business at all with whatever happens to them. He’s angry at her because she butts into his own business and talks about someone/something she doesn’t know anything about. A person who is irrelevant to him, trying to talk about what’s best for him is annoying.
She definitely isn’t mad at them for their relationship of being a student and teacher. You missed the point of why she’s trying to get between them. It’s because SHE LIKES HIM, she couldn’t care less if the person he likes is a teacher. She just doesn’t like the girl he has a crush on, and is using his crush’s position to have a chance with him.
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Don’t get me wrong- I think you chose the wrong character to try and pin the “they’re saying the right thing” on. If you said the older brother of the ML was trying to make them breakup because of morals- then yeah that makes sense. Because the older brother doesn’t have any hidden motives behind their breakup, he just doesn’t want them to get into trouble. But the girl that likes the ML is the one with hidden motives behind why she wants them to breakup, thus not making her a good example at all. She does not care one bit about a student x teacher relationship.

I agree that she's being nosy out of spite. But I didn't miss the point, you did. While the classmate is acting like that because she's jealous, it should still be a waking call for the older counterpart of this relationship to realise that she is in fact being immoral and possibly committing a felony. They shouldn't even be trying to defend their relationship because it's shameful.
I can't believe they're making her feel bad for thinking that a student-teacher realationship is bad. It IS bad, very bad.