Tell you what I think, for what it's worth... the church is faced with a basic dichotomy made even more complex by internal politics.
The dichotomy is the reconciliation of the real-world facts (in this case, about mental illness) coupled with the requirement to not violiate basic precepts of the religion. If the church ever took a public stance that challenged the existence of demons, they'd be challenging an article of faith as recorded in the new testament - Jesus's casting out of demons, giving power to his disciples over demons, etc.
Instead, they've decide to assume that "demons" exist, but very rarely possess people - and that the vast majority of people who think they're possessed are deluded or mentally ill. (Sounds like our applicants, no?) They can't simply say "There are no demons" or "Possession doesn't happen"... but they create hurdles about qualifications for demonic possession.
So, now to the politics of the matter.
Within the Catholic church priesthood and organizations, they have (like every other organization in the world) conservatives, liberals, progressives, fundamentalists, and so forth. The liberals and progressives pretty much defined the decisions at Vatican II, where things like making latin masses optional were decided, and formally exonerating the Jews for Christ's crucifixion was announced, etc. For some decades after that, the church was increasingly run by the liberal/progressive factions with the conservative/fundamentalists farmed out into the various parishes.
Over time, though, that's changed... and the conservatives and fundamentalists are slowly regaining their influence. As a result, several of the policies and "viewpoints" of the church - which are NOT part of the formal doctrine - are being affected. Possession appears to be one of those viewpoints, but it doesn't appear that there's any desire to return it to it's old format - instead, it appears that some elements are rebelling against "hiding it", and the church's position on the "frequency" of demonic possession.
There's a popular misconception that the RCC is a global empire of tightly controlled groups. The truth is that it's a global organization of semi-autonomous demenses, almost feudal in nature. Yes, there's some control over each diocese exercised by Rome through it's Archbishops, but when you get right down to it, the local dioceses' and parishs have an awful lot of independence.
Small wonder there's very little consistency with all this tug of war constantly going on...
