I really hope the situation with her sister is cleared up soon

I feel like the dad basically just scapegoated Olivia because he didn’t want the kids to realize it was actually his fault and blame him.

For real, because how was it the kid's fault that their father committed adultery warahek

My theory is that the real reason the duchess was divorcing him is that she found out about how he was orchestrating the abuse of Olivia. During the funeral scene he made a comment about how “she would never have known anything if you hadn’t met in the hallway”, but at that point she already knew about the infidelity, the existence of the affair child, where Olivia came from, and why the Duke brought her into the house (to take Estella’s place in the engagement). All really serious, but the only new thing she learned after meeting Olivia was about how she was being neglected and abused. And that is the moment the Duke referenced when he blamed Olivia for the divorce and accident.

The scene right before Olivia goes into the hall because she’s hungry shows a tray with a teapot and a single piece of moldy bread on the floor of her room.
1. Why would decent staff put a food tray on the floor when there’s plenty of furniture in the room.
2. She may have been brought moldy food to deliberately mess with her.
3. Even if she wasn’t eating due to emotional distress, a decent staff would remove and replace old food each meal, definitely before it could get that moldy.
4. If she was looking for food due to hunger #2 is much more likely.

She also learned that the staff and nanny deliberately degraded and manipulated her, probably to generate a sense of guilt and low self esteem intense enough that she would be an obedient puppet for the duchy. She only got involved with her when she found out about the abuse.