SB, personally, I have never believed that. But, even today, the majority of the GOP politicians would be happy to throw LGBT people under the bus. I think that majority with this opinion within the GOP is slowly decreasing and it will eventually not be acceptable even in the GOP. I have no idea how long that will actually take.
On the grassroots level, what majority opinion? Maybe try to stop some of the hate (I don't mean you personally as most are far more vitriolic than you could ever be) and I think you'll see change faster. Given the animus from the left toward the conservatives I think there should be a lot more hate right back, but I sure don't see it. And I guarantee you that I know far far more conservatives than you do.
True, maybe I see and know a different 'conservative' given that I'm in the upper Midwest in a city long known for acceptance and moderation. However, I find far more acceptance being gay within the conservative community versus conservative within the gay community.
Get involved, befriend and change instead of throwing stones from the other side.