They care, it's a big misunderstanding, I'm not talking with proof and spoiler on hand, just pure opinion.
I think they care, it's just she closed her heart and thought they hate her so she gave up on reaching out while the brothers can't find the right words, action and mind set to approach her without sounding and acting like jerks
Spoiler:
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I'm not sure if this already happened in the manga but:
Ok so this happened when they were children, she was baout 12. The pink haired brother didn't like her from the start and decided to get her into trouble. He took a neckless from his missing sisters room and then placed it in her room in order to make her look like a thief. She was confronted by the dad and the buttler said that he saw her sneaking around there, she told the truth (that she didn't do it) but the dad didn't believe her and thought she was a thief. I peronally believe this to be the reason as to why he (the dad) doesn't trust her. Atfer this he brought a jewler to the home.
I'm sure this hasn't happened in the manga yet:
The brown/black haired brother hated her because she came into the mansion as a replacment for his sister.
The dad was proud/happy because of what happened with the prience (her neck) and only cared about gaining an advantage against the royal family. He decided to reward her and not punish her. They did her her medical care when her neck got cut.
The servants don't like her at all. She doesn't have a body guard (someone in her position should have a few) and hasn't been allowed to leave the mansion except for official social gatherings, she was in a endless time out.
The game is rigged so that she is hated. It turns out that there are quiet a few things about the game she didn't know about, there were scenes she never saw when she played the game. Everything happens for a reason.
The prince is the meal lead. I think that the slave slodier guy she will get is a yandere.
As the story goes on the fmaily will realize their mistakes more and more.
Everyone will blame/assume that she tried to kill the prince. The pink haired brother will believe her cuz unlike other people he ain't a dick.
The games heroine (the sister) is basically evil/ in a cult.
27 pages of pure spoilers:
https://forum.novelupdates.com/threads/death-is-the-only-ending-for-the-villain.97486/
Check this out but set it to english:
https://m.blog.naver.com/PostView.nhn?blogId=akazihaeh&logNo=221753431857&categoryNo=9&proxyReferer=
Raws:
https://newtoki63.com/webtoon/2134795/%EC%95%85%EC%97%AD%EC%9D%98-%EC%97%94%EB%94%A9%EC%9D%80-%EC%A3%BD%EC%9D%8C%EB%BF%90-23%ED%99%94?toon=%EC%9D%BC%EB%B0%98%EC%9B%B9%ED%88%B0&spage=1
Indeed that's true though I still wouldn't put in in the caring category. There will be more developemnt over time but so far being outraged because someones neck got slit seems like normal human behaviour. The pink haired one is a lot nicer than the black haired one at the moment even though he is a tsundere.

These brothers clearly like her am I the only one seeing this?? They clearly do care for her.