Try #2. Hopefully this one fits the “rules” of this forum, PD: My first is not english, trying my best :)

As you said Latin America. I will focus this answer on everything below the Rio Grande :)

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TL;DR:

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" Christian evangelism was the ostensible motive for much of the early European interaction with the indigenous population of America. The religious orders of the Catholic Church were the front-line representatives of Western culture and the ones who met indigenous America " Cushner, 2008

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The evangelization of the Americas was not quick neither a sudden change. It was a process that undertook centuries, as it was basically superseding a myriad of system of beliefs with single shiny one!

Behind the First door:

Local populations(that survived the initial exchanges) found convenient(or were forced to) to convert to Catholicism in order to accommodate themselves to the new political panorama. Consequently, other inhabitants followed suit. It has to be noted, it was convenient for the Europeans that both worlds(the productive part of the new world at least) spoke the same system of belief (Christianity) and in Indo-European languages.

Behind the Second door:

Catholic missionaries where not shy with their (re)interpretation of the catholic dogma, more than often reinterpreting local beliefs into the catholic system of the belief. i.e. The many faces that Mary has in Latam like Virgen de Guadalupe o de Chiquinquirá. Therefore, facilitating the process in the first door.

Behind the Third door:

Eclesiastical institutions often represented "safe" harbors to the most brutal excess of the conquest and pre-Columbian political order. People slowly but steadily converted to Christianism around them. Thus, making easier the process behind the first doors. (Here I am not saying that the Church as an organization was not part of those violent excesses, tho)

Also, it has to be noted, that with few exceptions (such as the valle of Mexico or Marajó Island) the America that was found by XV century Europeans had really low population densities (why?, is a topic for another question (Cook, 2012)) compared to the panorama in Europe, the Middle East, India or China. This is important because it helps to explain why violence is not in any of the three big doors, as it would have been countless, pointless and cumbersome exercises with limited effects after a couple of years (Cushner, 2006). [Again here I am not saying that violence was not part of the road, i am arguing per se it was not]

Of course, there were specific cases of forced conversion trough the old plain sword method but I will argue those were isolated, and could not explain why today the region beliefs looks like they look. Also, as a Catholic myself, I suppose there has to be people that found the Gospel more compelling than their old system of beliefs, but as anthropologist myself, I do think the first three reasons explain better why my grans were "mas papistas que el papa" (harder popist than the pope)

LONG BLOCK OF TEXT AHEAD - the non quick answer

Before all we always have to remember that the Columbine Exchange was in essence that: an exchange, even if was lopsided, violent and dramatic. Different systems of belief that were isolated for thousands of years collided among political, economical and demographical upheaval (Nunn, 2012) , spiced by the pandemics that ravaged with the Native American populations . In that exchange there were agents in both sides that fostered the flows of information and material culture that shaped the human(an physical) geography of what it is today America. Language was not a problem, per se, as those agents make sure that both sides where able to communicate, as there was noted by the first chronist of the West Indies (Leon-Portilla, 1971). As there were imbalances of power, locals started using the colonial powers languages as lingua francas that allowed them to participate in flow of the Columbine Exchange . In simpler words, many times missionaries did not had to have deep knowledge of local languages as the communities have someone that translate for them. (Ordoñez, 2017)

So after everything I have said: If I was a native American in the XV century why in gods heaven should I abandon my system of belief?

FIRST DOOR - CONVENIENCE -

Systems of belief are sets of narratives that help a society and their constituents to explain and justify the state of the social and natural world in a given moment. XV century America was a stress test for many of those systems. Where the political and social order was interrupted by the apparition of the Europeans. This worked in many ways, in one way it was an opportunity for many elites(and people) to advance(or to carve a place) in the new social order of the young colonies (Cushner, 2006). Converting to Christianism was a move that helped Colonial elites interests, while advancing the new elites of the surging colonies. Furthermore, the societal advance of those who converted to Christianism, was something like a proof that move other to convert to Christianism afterwards. The olds gods were proved wrong (or proved their non-existence) into their eyes.

SECOND DOOR - SYNCRETISM-

European missionaries after the first shock, did their best to preserve in books and codices what they thought were the systems of belief of the people that they have to convert(and civilize in their eyes) (Watanabe, 1990). Why? Quickly they found that it was easier to re-signify the symbols and narratives of the peoples in order to ease the first process(and the forgetting of the old system) (Cushner, 2008) Therefore, Nahua goddesses transforms themselves into Virgin Mary personifications (see https://en.wikipedia.org/wiki/Our_Lady_of_Guadalupe), Old Festivals were tied to the Christian Calendar (See Día de los Muertos in Mexico o Velitas en Colombia), Andean origin narratives were linked to similar biblical narratives: Floods, Baptisms, Devil, Duendes we also have those here! South American folk Christianity still today lives with those additions where missionaries conceded native beliefs in order to speed up assimilation processes.

THIRD DOOR – SJW?-

Missions and their relatives: Ciudades Refugio, Bastiones, Conventos were seminal in what later became the American Nation States. Many American Cities from la Patagonia to Los Angeles still have their old mission names, in the same vein many cities in Colombia, Mexico or Brazil are called in old indigenous almost extinct languages preserved in original mission names. The conquest was brutal for many societies, and many times the social changes consequence of the former signified population displacements. The ecclesiastical authorities from the beginning of the conquest do recognized native American as soulful humans (see here for more of this wonderful XV century debate in Lopez Molina, 2015). Therefore, church settlements more than often represented harbors or places to be left alone from the worst parts of European extractives adventures. For the Europeans this was beneficial as well, as sedentary and village people were optimal for population and political control. (Walden, 2017) The demographical changes and the way how people settled were not a cause per se to became Christian, but it speed up the process of assimilation, convenience, syncretism and cultural hegemony (Cushner, 2006). Making a quick detour to the part of the question “there was no script” , well many times there was and the societies that did have script usually were those more stratified and population dense. Those societies, indeed, were the main effort of evangelization during XV and XIV centuries. While, hunter-gatherer or horticulturalist were left to their own until the North American Evangelical Missionaries of the XX century.