Conservatives have never been 'baying for the blood' of homosexuals en masse. If they had, then homosexuals would have been killed in bulk prior to 1960s, when social conservativism was much more of a cultural norm.
Yes, you can find fringe nutters who call for that, but you don't define a movement by fringe people, you define it by its stated ideals.
And the stated ideal of Christian conservatism in the US is as follows:
Homosexuality is immoral. Please don't hurt yourself like that, and if you're going to engage in it unrepentantly, you're not going to be allowed to be a member of a conservative church. Christians are called to Love our family, friends, and enemies, so we will still treat you with kindness though, if a sorrowful form.
That's the actual teachings of the Bible. Yes, you can say they weren't actually followed far too often, but that's the ideological core.
The idea that conservatives want to kill all homosexuals is, and always has been, nothing but left-wing fearmongering. Even when there were anti-sodomy laws on the books (which I think never should have been) in some states, imprisonment is not the same thing as killing, and by the latter half of the 20th century, those were being steadily removed, and rarely enforced where they still existed.
Even the cake baker, photographers, and pizzeria who've gotten fined or crucified on social media, all of them did not refuse service to homosexuals. They refused to cater/work at a homosexual wedding, because that would be seen as supporting something they believed was wrong. If just regular everyday services were asked for (the cake shop explicitly said 'if you want an already-made cake off the shelf, we'll sell it to you, we just won't make custom for a homosexual wedding'), they'd be happy to provide them.
The idea that a third to a half of the country is slobbering rabidly for the chance to 'kill the gays' is and always has been a myth in, at a minimum, post-1960 American culture and politics. It's a way to demonize the enemy to try to rustle up votes.
Don't believe me? Pick six random churches in your area out, call them up and ask the pastor what their stance on homosexuality is. See what kind of response you get. I'd be surprised if you got even one response that was more 'hostile' than 'It's unfortunate people succumb to sin and make it their identity' or something like that. If you got two without deliberately looking for known fringe groups, I'd be flat-out shocked.