Additionally, while it's not mental illness, there have been plenty of studies showing an inverse relationship between religiosity and intelligence (the more religious tend to be less intelligent).
https://pubmed.ncbi.nlm.nih.gov/23921675/
A meta-analysis of 63 studies showed a significant negative association between intelligence and religiosity. The association was stronger for college students and the general population than for participants younger than college age; it was also stronger for religious beliefs than religious behavior. For college students and the general population, means of weighted and unweighted correlations between intelligence and the strength of religious beliefs ranged from -.20 to -.25 (mean r = -.24). Three possible interpretations were discussed. First, intelligent people are less likely to conform and, thus, are more likely to resist religious dogma. Second, intelligent people tend to adopt an analytic (as opposed to intuitive) thinking style, which has been shown to undermine religious beliefs. Third, several functions of religiosity, including compensatory control, self-regulation, self-enhancement, and secure attachment, are also conferred by intelligence. Intelligent people may therefore have less need for religious beliefs and practices.
IMO, #2 makes the most sense and best aligns with my experiences with very religious people (fundamentalists). I can't tell you how many times, after interacting with a fundamentalist, I've thought to myself "Man, they are completely lacking in critical thinking skills!" Creationists are the poster child for that.
Regarding street preachers, a while ago a former street preacher wrote a blog post where he explained what motivated him to go out into crowded events, turn on a bullhorn, and shout insults at people (you're a sinner, you're damned, you're going to hell). According to him, the primary goal isn't actually to convert people; instead it's mostly about triggering people to yell, scream, and even physically confront you. Then when the day is over, you go back to your congregation, tell them about all the abuse you endured from the evil hoards, and then bask in the praise for "going into the lion's den" and emerging victorious. If you can throw in a few cases where you converted someone, even better!
That's why the best response is to simply ignore them, thereby depriving them of what they crave. I was at a tailgateing party one time and a street preacher showed up with his giant signs and bullhorn. He mostly hung out near the porta-potties and yelled at people while they waited in line. Early on, the tailgaters were pretty sparse, so the street preacher would approach people individually. When he came to me and started yelling and waving his sign, I just turned my back to him. He kept trying to get around me and get to my face, and I just kept turning and ignoring him. He very quickly gave up and moved on to the next guy, who started yelling back at him. As soon as the guy started yelling obscenities at him, the street preacher smiled and just amped up his rhetoric even more.
To me, that confirmed what I'd read. They really are trying to goad people into angry responses. But I don't know if I'd call that a mental illness. It's weird, kinda pathetic, and childish, but that's about it.