I think reaction to the Exorcist is very different depending on if your world view causes you to find it believable or complete baloney.
This is not so much a response to your post but I response to the idea that the exorcist depends on personal belief to be scary.I hear this a lot with the exorcist, but I don't know that it is so true. You don't have to believe in vampires or come from a culture where vampires exist to find them scary (just like I found horror movies grounded in Buddhist concepts, non-theistic ideas and folk beliefs outside my own scary if well executed---as any scary horror movie needs to be). I showed my wife the Exorcist, and she was raised Buddhist in a Buddhist country. She found it terrifying. I am not saying everyone will find The Exorcist scary. That is a very subjective thing. What scares me might not scare you.
But to go back to my original point, while the theology of the exorcist is important to its meaning, I don't think it is essential to the horror. Remember the possession part of the movie really begins with a secular stance, where people are dismissive of the idea of the demonic possession. It is about this girl who is experiencing alarming symptoms and much of the early movie is dedicated to her seeing doctors, having painful tests administered (in very realistic fashion for the viewer so you feel like you are undergoing the tests yourself---or at least present in the room), and exorcism is used as a last resort (and even the doctor doesn't really believe it is possession when he suggests it). So a lot of the horror is anchored in this experience of something terrible happening to a person and medicine not being able to find a cause (which happens all the time to people, as it can often take years to get a proper diagnosis). And the horror surrounding the possession is more about the falling away of a secular world view and a realization that the world is supernatural. I think there are a number of other ways to describe the movie. This is just how I have come to see it.
But I think people often just say it isn't scary anymore because that is something we see a lot on the internet. But most monster movies operate outside our present worldview (very few people believe in werewolves in the US for example, and you don't have to be a Puritan or believe in satanic witchcraft to find the Witch scary). It is almost more scary if you come in with a non-catholic worldview because the reveal of the film is that Catholic theology is the reality the characters are operating in. Because if you are Catholic you at least have some sense of the rules, but if you aren't, for example if you are watching it as my wife did, you don't know what the parameters of demon powers even are, and so there is a great deal of unknown. And if you are secular or an atheist (and I was in an atheist phase when I first saw the film and read the book) it is that dread that the world isn't as easily explained as you thought, that evil is real as a force in the world, that drives a lot of the reaction (at least it did for me). I don't find the idea that isn't scary anymore or is dependent on worldview, it to be more true of the Exorcist than any other horror movie in real life (I was terrified when I first saw Ju-On even though the take on Ghosts was way outside my cultural experience).
One thing I think is interesting about the exorcist if you watch it as part of the trilogy of faith, is in the first Exorcist, it is the presence of demons and possession that shake the faith of the characters (you can read the trilogy as Blatty almost playing out a crisis of faith), but by the time you get to the end of Exorcist III, it is the existence evil in the world that reinforces the characters faith, where George C Scott as Kinderman says in the climactic confrontation with Gemini Killer and "the Master" (possessing the body of Father Karras):
This I believe in... I believe in death. I believe in disease. I believe in injustice and inhumanity, torture and anger and hate... I believe in murder. I believe in pain. I believe in cruelty and infidelity. I believe in slime and stink and every crawling, putrid thing... every possible ugliness and corruption, you son of a bitch. I believe... in you.
The Master here is either the demon from the first movie or the Devil if I recall. What I like about this is how Kinderman is basically using the very thing that unsettled everyone's faith in the first film, to restore faith and defeat evil (this is somewhat interpretive as you have to presume Kinderman's thoughts after expressing belief in evil are that he therefore believes in God).