Well, the game is explicitly designed as a "dynasty simulator". Paradox advertises it as such, Wikipedia describes it as such, and it's specifically described how the goal is the glorification of your dynasty in everything from the screen that pops up when you start the game to the game over screen.
Caliphs and Fylkirs are interesting because they combine secular power with religious power. The Pope would be dull because he's a religious power but only a regional secular ruler--he's not the emperor of all Scandinavia or the padishah of Arabia, he's the count of Rome with a king-level title.
Not only that, but the church is already built up as more of an NPC than a player--the Pope's existence is currently to declare crusades, grant divorces, and with Sons of Abraham grant favors of money or claims to anyone who bribed him into power. Can you imagine how boring it would be to play him? You'd constantly be bombarded with requests to excommunicate so-and-so, to send such-and-such money, to give King John a claim on King Philipe's land, all entirely because of the opinion your character is calculated to have of other people, independent of what you as a player think of these NPCs.
I'm sorry if I come across as sarcastic or rude here, I don't mean to. I just think that cries to play a theocracy generally aren't really thought through from a gameplay point of view, they're more of a forbidden fruit situation where people want to play them because they're not allowed to.
(And it's already coded in to be able to play as nondynastic groups--IIRC, using console commands to play as the Pope, a knightly order, or a landed mercenary band reveals them to succeed the same way the Night's Watch does in AGOTmod.)