I think that a major difference probably lies in an incompatibility of terms/disciplines here. Conspiracy theories can be disproven easily using empirical, objective, material evidence based in science, history and reason. Religion - or God - cannot merely be proven or unproven by these streams - religion and science are not in conflict but simply lie within two differing ways of making sense of the world. You cannot disprove God through science but you can disprove CTs through science.
I am a Christian - you'd probably call me 'one of them' fervent ones - although if anyone ever associates me with the likes of WBC I might go a bit red in the face and subject you to half an hour's lecture on why that is markedly not so - so does my faith imply a mental illness of the same type as a CT? I know a couple of CTs, one Christian and one atheist, and both of them repeatedly ignore hard fact and evidence while asserting their views very loudly across social media. They back up their arguments with dodgy articles utterly baseless in any kind of academic structure of peer review and refuse to listen to anyone who points out fact. Now, you might well say that I as a person of faith do similar, but I and most of those who I know who are people of faith tend to be keen on the intersection of reason/science and faith and thus open to exploring the two together, and have found no reasons to throw out one for the other. The more I study of both, the more I remain convinced both are intertwined, and therefore my faith is not suspension of disbelief or a blind leap into something which can be tore down at the turn of a page. Theology is a discipline separate to yet not in opposition of science, just as is philosophy, psychology etc. Science is the study of the how, in general, and these others perhaps of the why as well as the how when it comes to matters of the mind.
I would argue that faith is not blind and unearthed to facts - when you have faith in a person it's because you've learned to have reason to trust them rather than because you simply want to. That's the same for me with things of God - I have reasons to trust and evidence based in a particular historical event rather than a general feeling or hope in order to feel I have a purpose or am in some way 'special'. Of course, my faith does make sense to me of meaning and purpose but it is not grounded in the need to have one, if you see what I mean.
I can't really speak for conspiracy theorists and what is behind their need to be right. I suspect it is similar to some more cultish expressions of 'faith' with their tendency to be more about control, though - perhaps a need to be more right than others, and perhaps for some born out of fear and a need to control what is around them as the world can feel out of control. I do understand the parallels you might make between that and faith/religion, but would argue that one comes from a place of fear and the other from a place of freedom.