I know Leoreth has said he's planning (at some point) to change the current religious system, so I'm just gonna say this now and see what other people have to say about it as well.
In my opinion, and as calad has said, if and when the Aztec, Maya, or Inca avoid the conquest, it would make most sense for those civs not to lose their native religions. There's no reason to assume that independent states of any of these civs would've adopted Catholicism, just as (someone may argue) there is no reason to assume that they would've not.
However, for many of the Indigenous peoples in Latin America, the religion they practice a mixture of Catholicism with their native religions, not actual ("Church-approved and dictated") Catholicism. Sometimes, the Catholic part is a mere façade and the religion is basically unchanged from what it was previous to the conquest. This leads me to favor the view that, if evangelization hadn't been supported by military force, it would've not been (as) successful and that any modern Aztec, Mayan or Incan civ would be more likely to hold to its traditional beliefs. I know basically nothing about Congo's history, but I would expect a similar situation (and if it's any guide, unless I'm mistaken, many of the descendants of the Yoruba people of Western Africa, after they were brought to the Americas as slaves and became evangelized, developed a Christian-looking form of the Yoruba religion, which we call Voodoo).
I strongly oppose the view that some religions (the belief system itself) are more important than others, while I accept that some religious institutions are better organized than others, but this is something that should be represented by the civics system. I think that either all religions (really all of them, including the so called "Pagan religions" which should be renamed at the very least to "Native" or "Indigenous") should offer the exact same advantages, or they should all offer distinct and balanced advantages (which I would prefer in the long term, since it requires more work, but offers more flavor and choices).
Anyway, thoughts on this?