The curia doesn't work, if the Papal State doesn't exist. Normally, when the Papal State is destroyed, which it is due to your converted CK2 save, another catholic theocracy becomes the Papal State. If none of those exist as well, the Papal State is gone forever and the curia is removed.
I presume, due to your save game conversion the trigger to remove the curia didn't fire or the trigger to move the papal state to another theocracy, if it exists, didn't fire. So whoever is curia controller now will remain so for all eternity. You may possibly destroy the curia, when you conquer whoever is curia controller now, but I'm not sure.
Edit regarding your edit:
Seems, the game somehow fixed itself. Possibly a catholic theocracy emerged somewhere and created the papacy or it's just something that happens with the converted. In any case 70 years is unnatural, the pope is usually very old, when he takes office already and a papacy rarely lasts for more than 20 years.
Regarding your question in accelerating the process, no. EU4 doesn't have the character level mechanics CK2 has. There is absolutely no way to influence another nations ruler or the popes demise except beating them in combat, when they choose to lead armies personally.