It actually goes pretty far in depth/darkness with her story. The story that I’ve gathered so far is that she is being abused by her father in always (mentally sexually physically) and in act 2/3 the mention this with natsuki’s poem about her father. Dropping coins maybe be an excuse for her to bend over to violate her. This is referenced later by yuri when she tells her to bend over and look for change near the vending machine, she breaks down and runs out of the room suggesting traumatic stress behind change. She’s also starved again in act 2/3 she passes out and has to be feed a protein bar because she’s too malnourished. Most of this can be confirmed using the wikia.
Viewing post in [SPOILER] Natsuki route
I don´t think it´s that Natsuki is malnourished though. In the first act, if you decide to help Natsuki during the festival. She states that Sunday that when you meet up with her that her dad MAKES her eat all of her food. But then again, she may be lying and refused to take the cupcake because she was scared her dad would find out she ate. I´m not sure. But, I don think she is being starved. I do like to here these conspiracies though!
Actually I think she is malnourished, though. She might eat all of her food because he usually never cooks. In the "Papa" poem, it says she likes when he cooks, which could mean he doesn't do it often. Also, it says she likes when he keeps food in the house. Plus, she is being abused (duh). Just my guess, though. I like your theory, though. Much more positive!