The major churches have also always had ways to circumvent the mainstream media cycle and relying on journos being nice enough to not cut out the inconvenient parts of what was said. Up to and including putting all the titles, names, quotes and page numbers in a booklet and handing it out to millions of people in case anyone wants to attempt to verify the experts. 99.9% of the people may be unwilling or unable to, but if anyone of the rest would, such info can spread in a community.
What major churches?
OK, I know you're not an American, but I'll explain. There was no major denomination of Christianity in the US that could have done this except the Southern Baptists, who did, but who's influence in the country was HIGHLY regional (mainly the US South and Appalachia). You'll ALSO note that in the South and Appalachia LGBT issues have consistently lagged behind the rest of the country. In part due to their influence.
Why this wasn't done elsewhere is due to that issue I brought up earlier, the Modernist/Fundamentalist split. The Modernists took over all the major organized Protestant denominations in the United States EXCEPT for the Southern Baptists (who fractured off the larger Baptist convention over, well... segregation and jim crow, no they were not on the right side of that issue). The remaining Fundamentalists effectively were scattered and ended up in continually splintering various conservative Christian protestant denominations that rarely grew to large scale. The biggest of these denominations you've probably not even heard of, as while that ARE actual denominations of Protestantism, they're all much smaller and with less influence than the old heydey of the "Mainline Protestant" Churches. What this means is that there wasn't any real central massive organized churches who could push publishing these things, and meanwhile the media would turn to "Christians" from Mainline Protestant sources who would claim, as they were part of the academic elite, these things were just made up and lies, and condemn the conservative Christians for lying about these things.
One pattern that can be seen is that the conservative position on abortion surviving better than the marriage position is not unique to USA, it's also fairly common in Europe, could be some kind of general western cultural pattern quirk.
I think it has to do with the underlying position on abortion being stronger and more readily accepted than the underlying posision on marriage.
The fight over the definition of marriage was lost as far back as the 19th century with the Romantic movement. Once marriage was seen as the culmination of romance and an expression thereof, rather than being about forming a family for the future focused on children, the underlying social logic for "marriage" was easily subverted to support same-sex relationships. After all, if two people love each other and want to commit to each other for the rest of their lives, why does it matter the sexes involved?
Abortion, on the other hand, is a lot harder to justify even under strong atheistic individualism because, well, you can't get away from the FACT that an unborn child is still a living human being, and killing other humans makes humans uncomfortable, especially for selfish reasons. Further this is one of those areas where the emphasis on material science in many respects BACKFIRED on the hedonistic libertines, as science improved the developing fetus became more and more humanized (to the point where pro-abortion people tried to suppress images of early human development because the fetus looked too human and they feared this would cause empathy for the fetus (which it does) and weaken the pro-abortion position (which it did)), thus people's recognition of those baby's inherent human right to life. It's not as extensive as it would be if the media was impartial or even actively anti-abortion (if the media was actively anti-abortion it would be banned everywhere and everyone would see it as a monstrous thing)... that society is as anti-abortion as it is is very telling how weak the actual arguments for abortion really are and I hope one day we'll remember abortion as an atrocity on par with slavery.
