regarding the question in the title: who pays for them... well they do; the labor they do produces A LOT of value for the USA. Like at the most basic level they act as the backbone for the agricultural business in the usa, without them the crops rot and food becomes scarce.
But in truth they generate vast ammounts of wealth beyond just that; thing is the wealth ends up in private hands, those of the big farm owners and big agrobusiness, and thus is untaxed and out of circulation, confined to interest generating vaults.
as to blaming the vatican, that misses the mark.
Firstly, the Roman Catholicism in the USA and in Latin America are fundamentally different beings. like the conceptualization you present is one that's very... well protestant. Like, for example, Biblical Literalism is unheard of in Lat Am (outside of protestant enclaves) and Tradcaths are not a thing (despite the more interwoven role that catholicism plays in the societies of the continent)
Secondly: Religion exerts soft power, it does not compel thoughts nor are the people of the global south some easily misled fools, blindly following the words of priests. (which is not to say that the church has not been used politically, to affect public sentiment, but then the institution is more of an instrument than a player) This ignores the divergence of goals and motives within orders of the church; Liberation Theology was (and is) a pretty big thing in Latin America; and it being denounced by Rome has more to do with which Great Power controlled the Vatican in the aftermath of WW2. (also painting it like its some issue wrt excessive latin american fecundity is verging on some weird eugenics shit that... just no)
The RC church was not the primary engine or benefactor of Colonization of the americas, it was a tool FOR colonization, but blaming the RC is like blaming the gavel for the sentencing, not the judge. The benefits from colonialism went to the Spanish crown and the HRE, and later to the various other Great Powers that sought to administrate and control the region.
Which brings us to the Monroe Doctrine and what the primary source of instability and violence in Latin America over the past 100+ years.
It's been the USA. It has continuously helped overthrow democratically elected governments, provided monetary and material support to counterrevolutionaries, crime syndicates, fascist terrorist groups, and isurrectionists in order to preserve its business's interests in the region (cf Dole), or combat the spread of "communism" (cf Chile), or as a part of its war on drugs (which is ironic as all hell because many of the cartels are the direct result of USA intervention, cf the Iran-Contra affair; also the link between the Zetas and the School of the Americas). And on top of that the gun policies of the USA ensure a steady flow southwards of arms.
Blaming the religious institution is nonsensical. Instead one ought turn to either the State that benefits from the crises in Latin America. There are Countries who benefit from having an underclass they can underpay for physical labor, there are Countries whose mining companies are able to move into Lat Am and compell usurous contracts as a result of the relative poverty of the region. There are Countries who field private "security" contractors to the aformentioned companies, benefiting from guvernamental impotence in order to privatize violence. And Countries whose banks demand Austerity from nations that need public spending.
Who pays for the migrants, they do; their home countries do. They pay in the work they do, they pay in the wealth extracted from their homes, and the safety stolen from them. And the USA (or rather the economic and political upper crust of the USA) are the ones who recieve those payments.
As to why Americans won't take those jobs... well for one, they do; prisioners are still American (and also there are a lot of desperate ppl that end up taking those jobs). But the reason the average american wont is because the job is physically devastating and almost entirely unregulated or even flagrantly violating to their legal rights (which is to say, the farmowners can't treat citizens in the way they treat migrant labor, citizens can fight back)