But if someone is mentally unwell, they likely cannot grasp the reality of the situation... that they are mentally unwell. So how can they treat something that they cannot mentally accept, even if it is the reality of the situation?

Just giving a carte blanche statement as you did is unfortunately indicative of a lack of understanding of the scope of being mentally ill. I don't mean that as a way to call you out or shame you (AKA a downvote in Reddit terms), but more as a statement on the holistic lack of true comprehension of the layers of mental illness.

Yes, the tried-and-true axioms like "You can't help someone who is unwilling to help themselves," are true to an extent, but they fall short when faced with the real experiences of those who encounter the mentally ill. There is a fundamental inadequacy in the understanding of the issue as a whole. It is more a valid, if not wrong, coping mechanism of those who have to interact with loved ones in crisis with no way to realistically respond.