I’m not a psychiatrist, but as a teenager I worked for an elderly woman who I later found out was a paranoid schizophrenic with organic brain decomposition. (As an aside, if you have any empathy at all, it’s impossible to get to know someone like this and not find their situation heartbreaking.) I agree that you cannot talk the mentally ill out of their delusions, hallucinations, etc. But this would only be relevant to the claim that all religious belief is mental illness if it were the case that all religious belief is the result of mental illness. But I don’t think all religious belief is the result of mental illness and I’ve never seen a convincing argument for why we should think otherwise.
For my part, I’m impressed by work in the cognitive science of religion which supports the idea that most humans have a Hyperactive Agency Detection Device (HADD). HADD explains why most people, especially most neurotypicals, have an overwhelming tendency to explain mysterious phenomena by appealing to invisible agents. It also explains why people on the Autism spectrum, who have varying degrees of mindblindness (and so to varying degrees are unaware of the beliefs, desires, and even (in severe cases) the existence of visible agents), are more likely than neurotypicals to be naturalists.
If that explanation (HADD) is correct, I wouldn’t call theistic belief a mental illness any more than I would call other types of cognitive biases a form of mental illness. Instead, if I were going to use labels at all, I would call supernatural belief the result of an often effective but imperfect cognitive mechanism, a mechanism which is the byproduct of blind evolution by natural selection.
Also, if it were the case that someone cannot be persuaded to change or give up entirely their religious beliefs, then we would expect that testimonies of converts and deconverts would make no mention of rational arguments. But that isn’t what we find. There are many people who became atheists because of something they read, whether it was Richard Dawkins’ GOD DELUSION, Bertrand Russell’s WHY I AM NOT A CHRISTIAN, or whatever.