(CNN) There's a common saying in Catholic charity circles: We don't help people because they're Catholic, we help them because we are.

To understand that sentiment is to understand why so many Catholics were incensed by accusations from a former Trump administration official on Thursday that the church has an ulterior motive in advocating on behalf of undocumented immigrants to the United States.

"The bishops have been terrible on this," Steve Bannon, who until recently was President Donald Trump's chief strategist, said in an interview with CBS's "60 Minutes." "You know why? Because, unable to come to grips with the problems in the church, they need illegal aliens. They need illegal aliens to fill the churches. ... They have an economic interest."

"Boy, that's a tough thing to say about your church," answered the interviewer, Charlie Rose, noting that Bannon himself is Catholic.

Bannon responded that on doctrinal issues, he respects Catholic leaders, naming Pope Francis and Cardinal Timothy Dolan of New York.

But immigration policy is "not about doctrine," Bannon said. "It's about the sovereignty of a nation. And in that regard, they're just another guy with an opinion."

Bannon's comments larded insult on top of injury, Catholic leaders said. On Tuesday, Trump announced the end of a program known as DACA that had protected from deportation nearly 800,000 young undocumented immigrants brought to the US as children. Many Catholic bishops strongly opposed the move, calling it "heartless" and "reprehensible"

Their reaction to Bannon's comments was just as unequivocal.

Catholic bishops, and particularly their American umbrella group, the US Conference of Catholic Bishops, like to position themselves above the partisan fray, rarely responding to political jabs. But in this case, they quickly counter-punched.

James Rogers, the USCCB's chief spokesman, called Bannon's comments "preposterous" and accused the conservative agitator of presenting a false choice between compassion and security.

"Our pro-immigration stance is based on fidelity to God's word and honors the American dream. For anyone to suggest that it is out of sordid motives of statistics or financial gain is outrageous and insulting."

In an interview on Sirius Radio, Dolan responded in kind. "I don't really care to go into what I think is a preposterous and rather insulting statement that the only reason we bishops care for immigrants is for the economic because we want to fill our churches and get more money."

But in one regard, Bannon is correct, Dolan said. Immigration is not necessarily a matter of Catholic doctrine. In some ways, it's deeper than that. It comes from the Bible and Jesus himself.

"And the Bible is so clear, so clear, that to treat the immigrant with dignity and respect, to make sure that society is just in its treatment of the immigrant is Biblical mandate," Dolan said.

"It's clear in the Old Testament -- my Jewish neighbors remind me of that all the time -- and it's clear from the lips of Jesus when he said, 'Whatever you do to the least of my brothers you do to me. When I was a stranger -- meaning an immigrant or a refugee -- you welcomed me.'"

What Bannon got right

Bannon was right about one thing: Latino immigrants are providing a lifeline to the Catholic Church in the United States.

As non-Latino white Catholics have left the church in droves during recent decades, Latino immigrants have flooded in to take their place. The turnover has produced tension in some parishes , but on the whole, Catholics have taken pains to welcome Latinos into their new, American congregations.

According to a recent study by the Pew Research Center, more than a quarter of American Catholics (27%) were born outside the country, mostly from the Americas; and one out of three American Catholics now identify as Latino or Hispanic.

That number is expected to grow: Christians are by far the majority of documented and undocumented immigrants to the United States, Pew has found. Of the 11 million undocumented immigrants living in the United States in 2011, for example, 82% were Christian, mostly from Latin America.

But Catholic bishops have never argued that Latino Catholics -- or any Christians -- should get special treatment from US immigration authorities. In fact, they've said the opposite.

When Trump announced in January that Christian refugees would be given priority under his administration, the US Conference of Catholic Bishops strongly disagreed.

"We believe in assisting all those who are vulnerable and freeing persecution, regardless of their religion. This includes Christians, as well as Yazidis and Shia Muslims from Syria, Rohingyas from Burma, and other religious minorities," said Bishop Joe S. Vásquez, chairman of the bishops' Committee on Migration.

"We need to protect all our brothers and sisters of all faiths, including Muslims, who have lost family, home, and country. They are children of God and are entitled to be treated with human dignity."

Immigrants' anxiety increasing

Monsignor Kevin Sullivan runs the New York branch of Catholic Charities USA, the church's large social service agency. The charity has offices that service Harlem, a predominantly black neighborhood where the majority of residents are likely not Catholic, he said, and Washington Heights, a largely Latino area where they likely are.

"If no immigrant ever entered the door of a church or ever gave us a dollar, our policies and our doctrine would be the same as ever: We believe in the core dignity of every human being, regardless of religion."

After he heard Bannon's comments on Thursday, Sullivan scampered down to the church's immigration office in lower Manhattan, where caseworkers help immigrants and refugees with interpreting English, legal issues, job training and placement, emergency assistance, counseling and childcare.

A few months ago, the monsignor said, an immigrant mother familiar to Catholic Charities caseworkers came into the office with a dramatically different appearance. She had dyed her hair blonde and altered her makeup, hoping to look less like an "immigrant."

"Among the people we work with, the anxiety is ratcheting up every day," Sullivan said.

At the same time, Sullivan said, caseworkers have been upset by the "non-reality based" rhetoric surrounding immigration and immigrants. On Thursday, the tried to boost staff morale by encouraging them to focus on two things: protecting young immigrants from deportation and lobbying Congress to finally pass immigration reform.

In the end, Sullivan is willing to cede Bannon's argument that, when it comes to immigration, Dolan and Pope Francis are just "another guy with an opinion."

"We're all just borrowing our opinions from another guy," Sullivan said. "And that guy is Jesus."