For some background on the LDS Church and race, see my answer here on the church and abolition.

I'm hesitant to say that no one in the LDS Church ever invoked the defense of "religious freedom" to justify its ban against blacks from holding the priesthood or entering LDS temples; some savvy Googling or thorough read through of speeches might turn up a Mormon leader somewhere using the phrase. But primarily, no, they did not use the argument of "religious freedom" to oppose Civil Rights, and there are some key differences between the LDS Church's reaction today to LGBTQ+ issues and the church's priesthood and temple ban from the 1850s to 1978.

First, the LDS Church found itself in the mainstream for most of its history when it came to race issues. There was segregation in Utah, and violence against blacks. Because the church and its members behaved mostly like other Americans, there was little attention given to it, except of course by black groups, who were largely ignored or maligned by whites. But when opposition began to gather against the church, particularly on college campuses, in the 1960s and into the '70s, the church's response was a confused one, stemming from different personalities in LDS leadership. President David O. McKay was cautious, deferential, and PR savvy. Apostle Ezra Taft Benson was a John Birch firebrand, happy to label Martin Luther King Jr. and the Civil Rights Movement as a Communist conspiracy. Apostle Hugh B. Brown was progressive, embarrassed by the ban, and lobbied for its reversal. A majority of LDS leaders were simultaneously not troubled by the ban and not openly opposed to the Civil Rights Movement. Therein lies the key difference between civil rights for African Americans and civil rights for LGBTQ+ people: one was an antiquated nineteenth-century racist doctrine Mormon leaders were stuck with, the other is a fight of their choosing. Leaders believed the ban had come from God and would have to be removed by God. Most were not active supporters of the Civil Rights Movement and they shared many of the same opinions as other white Americans, such as MLK as too aggressive, too far left, blacks too willing to demonize whites, etc. They saw themselves as not racist, but the other side as too radical.

In other words, there was a kind of compartmentalization going on with Mormon leaders. They did not see the rise of civil rights for blacks as an existential threat or as expressly against God's will (compare this to the LGBTQ+ movement). I don't want to overstate this because LDS leaders were elderly white men and they nearly all had the kind of prejudices in the 1950s, '60, and '70s you might imagine. Some of them were more vocal than others and some of them did invoke their theology and religious beliefs to express skepticism of the Civil Rights Movement. But it would be a mistake to picture the LDS Church in a unified opposition to Civil Rights in the '60s. And some official statements and some leaders expressed support for legislation and equality. But mostly the church's official policy was silence. They did not feel compelled to speak out, and they did not feel as though they were a moral force, one way or another, on the issue. For them, their own ban was not intended as a sign of support for segregation or extralegal violence in the South. As time went on in the 1970s, most leaders saw the ban's reversal as inevitable, especially after some of the "old guard" passed away, like Joseph Fielding Smith. Since then, the church grew embarrassed over its past ban and struggled to find a way to explain it. Some remained defiant: this ban came from God and the church has nothing to apologize for. Others though have sought more outreach with black leaders and black Mormons to try and understand how this happened. It's delicate for a church where most members believe that their leaders are, perhaps not infallible, but would never lead them astray over such important issues. The church most recently posted an essay to its website explaining the history of the ban and blaming it on racism, not on a revelation from God.

Contrast all of that with the church's response to LGBTQ+ civil rights. The church has been universally opposed to it and outspoken, even on matters that may not directly affect them. They saw it as a fundamental threat to civilization; perhaps not in the crude sense that humans won't be able to reproduce, but at least as a sign of such degeneracy and depravity and immoral permissiveness that humanity itself is under siege. They've tempered that kind of viewpoint, but only a small amount. They have opposed even basic, nearly universally agreed upon steps, such as a recent attempt to ban conversion therapy in Utah. Leaders fear being forced to marry gay couples in Mormon temples, something so visceral to them it might be likened to fears of miscegenation during the Jim Crow Era. People who watch the LDS Church today and follow this sort of thing are surprised by the intensity of their opposition to gay rights. For a church so often savvy with its PR, so often cautious in its statements, the exception is its opposition to LGBTQ+ rights. The botched rollout of the "November Policy" and the loss of what was likely 1,000s of members over that was, and remains, shocking to those of us specializing in Mormon studies. The church sees itself as an increasingly lone moral voice in the wilderness.

Where the LDS Church sought to sit on the sidelines and opted often for silence on black Civil Rights' issues, it has remained vocal on LGBTQ+ issues. And where opposition to black Civil Rights was often linked to religion, but not exclusively religious, opposition to LGBTQ+ issues is almost universally driven by religious groups in the United States. That has led to the rise of the "religious freedom" argument. The church has pushed this argument strongly because the United States, as well as other parts of the world, has seen such an abrupt reversal in attitudes over gay rights and gay marriage. The LDS Church has few allies left in its fight, and now it has retreated to something of a defensive posture, and religious freedom is an effective argument for them to try and keep going. By all accounts, when protests erupted against the church in the wake of the passage of Proposition 8 in California, Mormon leaders were caught off guard and shocked at the reaction. They are better at speaking more moderately and have grown savvier (including some support for legislation granting some limited rights for gays), but looking at amicus brief court filings shows the depths of the church's opposition. They recently supported firing LGBTQ+ people in stark language that said hiring those people to work at the church would threaten the faith's values. In other words, even sitting in a cubicle next to a gay person was threatening.

Finally, unlike the discomfort many LDS leaders showed during the Civil Rights Movement and the fight over the priesthood and temple ban in the 1970s, LDS leaders are, by and large, fiercely proud of their outspoken opposition to LGBTQ+ rights.

For some good reading, see Max Perry Mueller, Race and the Making of the Mormon People (Chapel Hill: University of North Carolina Press, 2017); Matthew L. Harris and Newell G. Bringhurst, The Mormon Church and Blacks: A Documentary History (Urbana: University of Illinois Press, 2015); Gregory A. Prince and Wm. Wright, David O. McKay and the Rise of Modern Mormonism (Salt Lake City: University of Utah Press, 2005; Gregory A. Prince, Gay Rights and the Mormon Church: Intended Actions, Unintended Consequences (Salt Lake City: University of Utah Press, 2019).