I am not a health professional or counsellor, but here is my take on this.
I think there are two separate strands here.
i) Anything which affects someone's ability to function in their everyday life is affecting their mental health. Your sister does seem to need support to deal with the changes in her behaviour and the effect it is having on her relationships.
ii) The truth or falsity of the conspiracy theories are a separate issue. There are many of us, I'm sure, who believe there is some truth in certain things branded as "conspiracy theories".
Just because her mental health has been affected by her obsession, does not mean that all of these conspiracy theories are false. Some of her anxiety may be based on real dangers in our modern so-called democracies.
If family members try to help your sister by telling her that these theories are all false, I doubt that will help - it doesn't sound as though that is what you are trying to do though.
She probably needs someone to talk to her in depth, acknowledge that there could be truth in at least some of the theories she believes, but point out that she needs to be able to cope with the anxiety these are causing so that she can live her life and not put close relationships in danger. On that basis, would she consider seeing a counsellor to help her take control of her anxieties and find strategies to help her cope. You may have examples of people you know who have been obsessed with things that may not be bad/wrong in themselves (sport, online games, working long hours) but have damaged their relationships etc. Pointing this out may help her see it's not necessarily that her views are wrong or that she is wrong to have any anxiety over them, but it is more that she needs to be able to live her life in spite of these things and not be overwhelmed by them. Maybe even acknowledge to her that there have always been injustices, cruelty and cover-ups of various sorts in the world, but she shouldn't let the people who do these things make her so angry/anxious that it spoils her life and that of her family. She can't solve it all, but could take constructive action: she can maybe sign petitions, contribute to charities that fight injustice, join a well-established group such as amnesty international or something along these lines. Alongside that, she could get professional support to keep things in proportion and cope with everyday life.
On a personal note - until I was in my forties, I probably thought western democracies on the whole had professional, experienced people with integrity in politics, civil service and security services and films/books about cover-ups and people being killed to keep secrets etc was mostly fantasy or at least exaggerated. I am now firmly of the belief that the truth is probably worse than many of these works of fiction. I don't believe every "theory" but do believe there is very good evidence for certain people being murdered to prevent them exposing the truth and for the existence and cover-up of paedophile rings involving very prominent people and so on. I think there is a very real threat to democracy in the UK and US - we are becoming more like dictatorships. I am very worried about the future- especially for my grandchildren. That being said, I am determined not to be overwhelmed by it and know that my faith in God and the love within my family help me to try to live each day with as much peace, joy, love and compassion as possible in spite of these worries.