It’s pretty clear that the “she made a student cry & caused her to stop attending at her previous school” rumor that keeps popping up simply means the teacher was confessed to by a female student, turned it down, & the girl cried & stopped going to school out of heartbreak.
That’s probably why she’s become overly nice & sensitive to the emotions of others (like when she agreed to share the umbrella even though she was initially gonna reject it). In a way, she’s just like how the MC was(is), always trying to placate or slowly simmer the expectations of others—in her case, the expectations of kindness & attention people (girls) want from her. Like the MC whose reputation of a perfectionist, academically-gifted helper are indeed aspects of herself, they aren’t all she truly wants to be, show, or entirely be known for, & hinder her ability to be herself & build friendships. I wouldn’t be surprised if the teacher saw a part of herself in the MC at the start of the story.
All in all, not even touching on her other aspects (it’ll be peeling onions, layers), the teacher’s a pretty complex character under the surface with a dope character design, looks & personality wise.