The Catholic Church is an excellent analogy, I think. If you believe the polls, lay Catholics are more likely than the average American to support LGBTQ equality. They're more likely than the average American to reject discrimination based on religion.
Yet, as you point out, Catholic leadership in the United States is (on the whole with scattered exceptions) intensely homophobic and transphobic. My God, the US Conference of Catholic Bishops actually lobbied against the federal suicide hotline on the grounds that it's morally wrong to reach out specifically to LGBTQ people. They wanted Congress to remove that part of the program.
The National Catholic Reporter (an influential Catholic newspaper run and funded by lay Catholics) found out about the lobbying, and broke the story in a tone of genuine shock and horror.
They sternly morally reprimanded the bishops in a reverse of ordinary Catholic hierarchy. Lay people were scolding bishops who were acting in a way that almost all Americans (and certainly lay Catholic Americans) found shocking and unquestionably morally unacceptable.
And then ... nothing happened.
A few months later, the secretary general of the Bishop's Conference was outed as a gay man who constantly used Grindr for sexual assignations with other men.
He'd been going to expensive gay sex clubs on the Church dime while spearheading the effort to lobby Congress not to provide suicide support services for gay and trans people.
You can't make this stuff up.
And still ... nothing happened.
Pope Francis continues to make nice noises come out of his mouth about transgender and gay people sometimes, while continuing to run a Church that condemns us as disordered and literally depraved, as "destroyers of nature." (That last bit came out of Francis's mouth personally while he was on a trip to Poland.)
And then there's the fact that Francis personally signed a CDF responsum ad dubium instructing priests not to bless same-sex couples because God cannot bless sin. (Popes normally do not sign CDF responsa, so Francis had to be personally involved.)
All that to say that the vast majority of lay Catholics in the United States who are supportive of LGBTQ people and equality nonetheless accept oppression and persecution their leaders commit.
I mean sure, the pews are emptying out steadily. In part, that IS a response from lay Catholics on the LGBTQ issue. But mostly, the faithful are quiet. Mostly, the faithful quietly accept the extreme moral depravity of episcopal leadership – by not acting, by not withdrawing financial support, by not standing up to demand change even when leaders like the Bishop of Indianapolis engage in anti-queer witch hunts up to and including threatening to expel Catholic students from school for opposing his witch hunts in public. (I'm thinking of a case where the bishop successfully gagged a 17-year-old student by threatening to expel him just months before he was scheduled to graduate, which would have ruined his college prospects.)
I don't really get that. I don't understand how lay Catholics can be so quiet in the face of so much episcopal moral depravity.
Anyway, looking at Rand's data, I do worry as much as I hope. I feel good that such large percentages of Republicans say they support queer people, women, etc.
But ...
Will they be any more prone to act than lay Catholics? I don't know. At the moment, it sure isn't looking like it.
I don't know how to answer your question. What comes next? I don't know. I know this much: All it takes for evil to prevail is for a few good people to do nothing.