What if, and I mean what if, this game traveled through more than one timeline?I feel like this is the most problematic of the ideas you posted because of one simple thing:
Lizzy is enrolled in the school and people do not treat her like the new kid at any point.
Any time travel would have to occur well before the start of the game (and thus the school year) so that Lizzie could get entered into the schools records and legit be attending there (getting grades, doing projects etc) if she was from 1999 as an adult/teen (or any time other than the year she was really born in story) she would then have to get fake documents to enroll into the school (she would also need an adult to enroll her). Well what if Riley or someone hacked into the school and made sure it looked official even though she started this year, but then you run into the issue of nobody ever a single time treating her as the new kid most importantly Brad does not treat her as a new kid. Also when she says she dated a guy and looks at MC is she looking at the camera (we see the scene from MC pov) or is the mc off to the side and she faces off to the side when she says it?
That means if she really is from the past or whatever Brad has to be in on it which means he is helping us with time travel adjacent issues while knowing they are from time traveling but he slowly solves them using modern tech? Even though people he goes to school with are being threatened and blackmailed?
Brad the dude who is openly talking in class about his quest for a mlf harem (to the point that the teachers have to tell him to shush) is also a key piece in keeping the secret from MC? (his brother in harem questing??!?!?!? Not bloody likely!
Like I mentioned before, this isn't the first time the MC has saved Tiffany at the school, which means that the MC of the future timeline saved Tiffany before the MC even experienced the time he traveled through, and if I remember correctly, Liz transferred to the school a year ago?
We can make an assumption that the controller in the Mystic's hand can pinpoint the time traveled, and this will definitely be in the MC's hand in the future, so is it possible that the MC could have traveled to a timeline a few years after 1999 if he used it incorrectly the first time? Maybe meet Liz at that time?