It's been so long, I don't even remember why they're all fighting so hard to win her favour
Because she is a extremely precious person.
Someone who is so kind and caring about others, it is arguably to a point of fault, especially in caring for herself. And she has lived a extremely brutal and tough life so prior, so they both just want to make her happy and smile.
Meanwhile she is also naive and oblivious to people's motives and true nature's, thus she also invokes a protective instinct.
As for why they are actually fighting each other over her, is generally just a kind of sibling rivalry.(siblings rib each other and fight each other all the time) Both(all including father), want to be her favorite and be the most special to her.
As for super duper specifics of how they got there:
- Brother discovers he has a long lost sister. This sister has had to live in squalor and poverty for her entire life after the death of her mother, instead of being raised as the non royal princess she should've been. This sister has also been terribly maimed and scarred over the years to the point it is worse than the entire Duke household combined, only by some miracle is her face untouched. Furthermore this sister is the most kind, selfless person he has ever met who doesn't have a ounce of ulterior motives. Even better, his sister is a very powerful sword after and worthy of admiration. All in all she elicits extreme sympathy for her past life, a desire to protect her for her future, and a desire to make up for lost time and earn her smile.
- Pink sister has lived with serious bedridden illness for more than a decade. And watched her sister time and time again nearly get herself killed to provide her life sustaining medicine and food. While carrying that burden entirely herself, and concealing the lengths she goes through. Big sister is literally her everything, her light and salvation in this tough world. Now she has been adopted by big sis family, and been cured of her illness, she wants to repay her sister for everything and make her live a happy life.
- Their conflict comes from pink sister being the third wheel to the sister, an obstacle to the brother getting closer. While the brother is a threat to the pink sisters previous monopoly of her big sis. They have personalities that also get along about as much as cats and dogs. They both want to be the sisters favorite.
Or the two sentence version: MC is a extremely precious and innocent person who has gone through terrible hardship. The siblings both want the best for their precious sister and both want to be her favorite, thus they are siscon rivals.