And even if you think I'm wrong about everything else, there's no doubt we wouldn't have anything resembling the world's current persecution of LGBT people without religious justification.
Homosexuality was harshly persecuted under the Soviets and Nazi Germany, completely unrelated to religion. There's historical precedent.
I honestly don't believe religion has anything to do with it. I think it's authoritarian culture. An authoritarian leadership will want a homogenous population that's easy to understand and control. Those living in an authoritarian culture will be eager to attach their prejudices to authoritarian structure's willingness to persecute in pursuit of homogeny. The authoritarian structure does not have to be religious, and in the absence of religion, I firmly believe that social reactionaries will simply create alternate authoritarian structures to buddy up with others who share their prejudices and pursue their goals.
The one advantage that religion has is it assigns responsibility for the persecution to an authority that is above humanity and doesn't have to explain itself. So when challenged, the religious can simply say "because god says so", which is a lot easier to treat as a dismissive final statement than "because glorious human mortal leader who can be asked his reasoning directly and lives in that palace over there says so".
Yeah, it's no comfort to victims to assert that freeing them from persecution isn't as simple as eliminating a group, but I don't believe in false comforts.
Just like you said, it's not a real solution until the real problem is dealt with. I don't think that eradicating religion would be dealing with the real problem, and is a logic that could be argued to share common roots with its enemy. Worse, it gets in the way of dealing with the real problem, because it feeds religious people's persecution complexes, which usually makes them double-down on their fundamentalism. So we get more conflict and hatred over a pursuit that won't actually accomplish anything.