I have an older sister who is completely irresponsible with money, and has never done a thing to take care of herself or pay her bills. She also wanted to "be happy" in her work. She has a university education and absolutely no impairments or disabilities that would have prevented her from working to support herself.
To the best of my knowledge she currently lives (in another state) in government supported housing, and receives all sorts of tax-payer benefits re: food, medical care, etc. I really don't know, because I long ago realized that the old adage is true - "you can lead a horse to water, but you cannot make them drink."
I would never in a million years do the same thing in your shoes, and I doubt that very many people would. Your sister is an adult, and as such it is not your responsibility to pay her bills or try to control the choices she makes now.
I will hasten to add, from my vantage point as a cancer survivor, the only thing she needs to focus on right now is getting better. Frankly, worrying about whether or not a surgery is covered at this point is beside the point. Is it going to save her life? Is she supposed to decline to have it if it can save her life, but isn't covered?
Back when I got my cancer diagnosis, it was all I could do to remember to BREATHE for the next many weeks/months. Your sister is terrified right now, believe me. You can either be her big brother and provide emotional support as she goes through this, or you can be the guy who is "always right," try to take control of the whole situation, and thereby alienate her and your mother at a time when they are already stressed beyond belief with fear and worry.
The last thing I will say is that I cannot wrap my head around the fact that her medical plan actually provided you with her plan information. That is so far over the line, and breaks so many boundaries, it must be unethical, if not downright illegal for them to do that.