I'm actually in the opposite boat - I'm not even sure actual time travel IS occurring.
There are parts where the MC mentions that things don't quite match up to what he remembers (Isabelle and Jacklyn being there, Ms. L dressing sexy, Lindsey being clumsy, etc). So this isn't just a rewind, but a complete rewrite of his past even before he starts changing things.
The vibe I'm picking up is that this is some sort of Purgatory scenario and God/an angel is fighting against the Devil/a demon for control over your soul, in a sort of recreation of your past cobbled together out of your own memories and a few extra elements. Mrs. L pretty much screams some sort of infernal shenanigans (both rewards she gives you are described as being unnaturally warm, the ad for her exercise tapes is almost completely made up of "evil" references, Mrs. L could easily be short for Mrs. L(ucifer), and Lichtenstein could easily be a mash-up of Lightbearer and Morgenstern, both names for Lucifer in myth), and her being the Devil/a demon paints the conversation you overhear from the locker room in an entirely new light (the good force is basically accusing her of cheating at their game). The good force is pretty obviously the mystery person texting you and telling you to follow your heart, and could either be one of the other characters we've already seen, or one of the ones that have been mentioned but we've never seen yet (my money is on the janitor).
It could also explain why the game is keeping track of Love and Lust as separate variables - give in to lust and you're basically playing into the Devil's hands, but "listen to your heart" and pursue love and you're redeeming yourself.
To go a step farther, Isabelle and Kate would seem to be the main champions/tools of each side, with Isabelle always talking about making the world a better place and encouraging you to be better, and Kate... well, being Kate.
It's hard to say whether all the other characters are just figments of your imagination, angels and demons taking on the roles of the characters they're pretending to be, or if they're the actual souls of the people they seem to be, but all are possible...