I never could understand the point of having demons in the first place. What purpose do they serve in the Grand Design? To just go around possessing people?
And it seems rather pointless, since the demons give themselves away with the projectile vomiting, the screaming, the flailing, and turning their heads all the way around.
There are a couple of interesting points here since demons do serve a function as cultural constructs. The mental illness hypothesis is one way of viewing it, that people didn't fully understand the reasons for aberrant behaviour and so tried to explain it as having inhuman origins. I think there's probably some truth to this but not in all cases. Some cultures were aware that these behaviours were the result of sickness but often had a very limited understanding of the nature of that sickness.
Beyond that, demons and other evil spirits very often served as cautionary tales. Spirits were often associated with wild places, the forests, deserts, mountains and rivers, which were often perilous to travel to. To give an example, there are many folkloric examples of river spirits who would lure travellers into the water and drown them. The cautionary element here is fairly obvious, "Be careful around water." Jenny Greenteeth was one such bogeyman figure, apparently designed to frighten children away from playing by the water's edge.
Then there's the possibility that some of them helped people cope with an uncaring world. For example, some demons were blamed for causing miscarriages, stillborn infants or the deaths of mothers during childbirth. It makes sense to me that a scapegoat might serve as a coping mechanism when faced with a tragedy like that. The fact that demon folklore of that sort invariably has accompanying charms and prayers to ward such demons off suggests to me that this may be an attempt to exert some kind of control over a situation that made them vulnerable.
Finally, there's the storytelling element. People tend to think in narrative terms and a good story requires adversity. Demons, monsters and various evil spirits make good antagonists who are powerful, deadly and irredeemably evil. It's not always clear how literally many of folklore's monsters were believed in. It's also not clear how many who were literally believed in were originally intended to be viewed that way.
I'll throw out a disclaimer here that this is a mixture of both scholarly theories and my own conjecture. It's a complex subject with a lot of disagreement and the hard truth is that we may never fully understand the origins and intent behind many of these beings.
You'd think that they'd try to play it cool - possess someone famous or powerful - and try to act like that person, while subtly shaping the course of human events.
This part immediately made me think of the conspiracy theories along the lines of David Icke's reptilians and the believers in a Satanic New World Order. Suffice it to say that there are plenty of people who believe pretty much exactly what you said here. Their view is that there's inhuman influence at the very top levels of society.