I think the main issue with the supernatural stuff in this game is that you fail to establish this element early on and the plot just out of nowhere does a 180°, deciding it's now not a "realistic" school sim anymore but is having fantastical elements. And yes the quotation marks doing some really heavy lifting there.
If you want a suggestion:
I'd have a the intro start in the MC's childhood (~age 12) when he does some ritual that summons Lilith during which she implants a seed of her powers "which is nurtured by the fall of those around you" (to establish that "more corruption" = "more powers") into him and announces him her "future champion" (to establish that he doesn't fuck around as a minor which is an implication I'd rather avoid).
After coming of age he'd start using the powers, spreading corruption - and his cum - basically wherever he walked but I'd skip over that at this point to not make the intro drag.
As he becomes older and eventually a teacher he starts to use those powers both less and when he'd do so in a more subtle way to motivate, Lilith being not happy about the distinct lack of classroom orgies she hoped for uses her influence over you and your students to remedy that situation - which will be the reason for your firing.
And lastly you could have him start having dreams about Lilith - now calling him her champion instead of future champion - at the start of the game in his first few nights at the academy, so that when she finally shows up in the flesh the first time as you proceeded far enough with one of the girls he realizes that she is real and it weren't mere dreams, but him not quite making the connection that it was her who ruined his life or for quite how long he actually was in contact with her, which I'd slowly sprinkle in, showng how by the time he was 18 he obviously forgot about all the ritual etc, Lilith spectating from the shadows as he spreads corruption around his college and private life, using her gifts subconsciously and spreading corruption around without really knowing what he is doing and continuing to do so as he grew older with Lilith being pleased with herself and your progress until finally him realizing that it was her who got you fired - maybe that last part only on the good path tho as a final piece of motivation to get rid of her.