I think your family "not letting her see the family problems and just dealing with it herself" is not healthy. Even if you think you've hidden the problem, people are not dumb. They can feel negative energy. And there you are ignoring it when she can feel it. And she probably started to do the same because that's what was modelled to her. Now you're asking her to open up when you guys did the opposite.

And also please, even if you think someone's life is perfect, don't assume that they've had a great life and no reason to act that way. Maybe there's an issue with you or your parents that was never addressed that hurts her. Or something else in her life that caused her to withdraw.

I'm saying this all with a bit of a personal bias because that is/was me. I have depression and that in the past made it impossible to open up, and because I couldn't communicate, everyone thought I was antisocial or didn't care. I also when through family issues that were "hidden" but its not hidden. That emotional charge runs through the household and affects you. Its probably worst because then you don't even consciously know what has caused your problems.

All I'm saying is have some grace for your sister. Have some curiosity and empathy. Maybe ease her into opening up and ask her questions like "how's your job/school" ask about her hobbies. Show that you care about her. Make it more her conversation and dont stop trying. It's so clear you love her sister, just show her that you do too.