My general belief is that it just isn't relevant. My mental illness, for example, doesn't influence my own attitudes toward Apollo. Heck, it's entirely possible that some of these people are mentally ill--but it's depression or something, and it has nothing to do with their attitudes toward Apollo. Mental illness comes in a wide variety of appearances, and most of them wouldn't cause the things we see here. To be honest, the only people I believe to be mentally ill with whom we've dealt, I have generally believed to be quite seriously ill, and it has made me regret that there is nothing that can be done for them.
This is my real problem with the speculation about about mental illness and conspiracy theorists. It is seldom from a place of compassion. It's usually mockery. It's usually a reason to dismiss outright anything they say, as if the mentally ill never say anything of value. And if they do remember that I am mentally ill, I'm not supposed to take it personally, because I'm not one of those people. Certainly true, but it's not because I'm bipolar and they're something else, not that most of the people speculating even know enough to know what illness they're speculating about.