So I have bit of a mini theory and I know not many will see this so I'll be posting this later somewhere else but if someone does see this, feel free to improve on it.
So upon browsing the Project Libitina website which is supplied by a Sayori Pic (Check the Easter Egg page for more info), you get this report which may hint at possible experimenting but what interested me was the listing of a "Third Eye Activation Test" and a "Third Eye Suppression Test" which seems weird but with context, it makes sense. I mean, a Third Eye can refer to a gateway to a state of higher thinking which usually means psychic powers, something Monika displays through out the game on any given run and is also hinted at that Sayori can be near this "Meta-Awareness level" (If the trivia section for her page is to be believed). Based on this information, I think the club, or maybe even the school, can be a cover up for some government or company experiments which is being done on these girls.
Although this theory does explain some stuff like Monika's "abilities", I feel like this is still half baked or even based on a dead lead. It can be used to explain Monika's self awareness, Sayori's depression (Experiments can do that), the abuse of Natsuki from her father (If we were to say that her father is actually just a scientist pretending to be her father) and Yuri's obsession with self-harm but I on a reread, somethings on the report fits doesn't fit Monika's character. I mean, there is a listing for self-harming which is a Yuki thing but there's no proof either denying or confirming the fact that Monika yet.
Just tell me what you think on it so far because I will be polishing this for later
EDIT: Here's a link to a more complete version of this theory that I just posted on Reddit and it now includes more theories