This is why I'm especially skeptical of her assertion that is "has nothing to do with you." It _does_, because if/when she needs help, of course you're going to want to be there. Still I can't think of a way to say "It is my business because it's possible that one day you will have dementia and I'll have to pay for someone to care for you all day."
Sure, the assertion is fallacious. Pointing that out doesn't actually change anything, though. You can't force intimacy (and sharing otherwise private/sensitive information can definitely qualify if you feel that way) where it's not welcome. The 'problem' (if you see it as one) is that sdrothrock's mother doesn't see them as an ally on this front. That's what'd need to change.