It's not manga logic and she doesn't have to be a teacher. In most elite schools in East Asia, especially in Japan, alumni hold disproportionate power over the student body due to the influence of alumni associations. To get a feel of this influence, go read medical and business manga where the reach and extent of alumni associations are discussed in detail. It's not fictional. Suffice to say alumni associations open doors for the students. Be it entry to elite corporations, access to prestigious hospitals, or even something as simple as a comfortable stay in a university (where a lot of your high school alumni can be found), your graduated seniors are important for your future career.
Conversely, it is important for said alumni to maintain the quality of their juniors. If you're expected to hire your juniors as underlings in your personal clique in the company, you damn well make sure they're competent and presentable. This usually means going back to school to interact (intervene?) directly with the students. I went to an elite all-male high school and my first job was working as underling for an alumnus of that school. Literally all the top graders of my high school graduation class got recruited there (I wasn't one of them, just knew one who was). Heck, I got my second job due to the boss having unshakable belief in the quality of my school's students. As you can see from the fact I'm even commenting here, that belief may have been misplaced, but, hey, not my fault.
For the record I don't mess with the school or its students. I know guys who do, but personally I dislike this system (yes, despite having taken advantage of it twice).
If anything, big sis' LACK of influence in the manga is severely unrealistic. At her age my aunt was complaining loudly to school administrators over the perceived loosening of students' "morality" in the years since she graduated (from her all-girls elite Catholic school). The scary thing was that they listened to her and let her literally retrain the school's student council on how to do their jobs. This is an old family story (happened in the early '70s) told proudly by mom about 20 years ago to my decidedly unenthusiastic little sister who at that very moment found out the source of her high school woes...