The motivation is mostly religious.
Homosexuality must be a choice, otherwise God created homosexuals as they are, and homosexuality is just another part of Nature.
But if that's true, homosexuality can't be a sin, because sin is always a choice.
But homosexuality is a sin because the Bible says it is (even though it doesn't really,) therefore it must be a choice.
P: Sex outside of marriage is a sin.
Q: Marriage is between a man and a woman.
Therefore [P and Q] implies that homosexual activity is wrong but the mere act of being attracted to the same sex is not inherently sinful.
As for why that should influence our votes for public policy, the reasoning goes that sin always brings trouble, death, and non-well-being to society, therefore influencing public policy away from sinful societal patterns and acts is one important way to love one’s neighbors, since it will result in greater well-being for them.
Also, why do you feel the Christian sacred texts doesn't really condemn homosexuality? I feel you have to do a lot of extra-textual analysis and theory-crafting to reach that perspective, so it's no surprise that Christians in general don't buy that idea.
And... it also condemns masturbation, marrying foreigners, eating fish, letting menstruating women inside the city limits and mixing fabrics.
Someone living in Biblical times probably wouldn't have any concept of someone being "straight" or "gay," just of someone following cultural norms or violating cultural taboos. So to say the Bible condemns homosexuality is a misappropriation of the intent of the text outside of the cultural and temporal context to which it was intended to apply.
This comment you're reading has a dead sibling that pointed out the commonly-held Christian doctrine that some of the Bible's laws don't apply to modern Christians.
The commenter is absolutely correct, and here is a link from a famous and well-regarded theologian on the subject (R.C Sproul).
https://www.ligonier.org/learn/articles/which-laws-apply/
In summary, there are civil, ceremonial, and moral laws in the Old Testament. For example, "no murder" is a moral law, while dietary restrictions are a ceremonial law.
I'm not commenting on the validity or "correctness" of this doctrine, as I am not a theologian. I just think it's worth confirming that this is something a lot of Christians believe.
In general, NT has plenty of stuff that can be appropriated for fire-and-brimstone preaching. Most of it is Paul's, which is why some consider Pauline Christianity to be a mutilation of what was preached before.
[0] never met Christ, so he had to pretend he had some sort of vision to get in the club... Jesus treated the money changers better than He would have treated Paul.
There is a very convenient Christian doctrine where there are "three types of law in the Bible", which explains that some rules are meant for Israelites and some are meant for everyone (and you [or the church] basically get to guess which apply to you).
It always read to me that it was more about turning men into women than it was about actually performing homosexual acts.
When you say "homosexuality" do you mean homosexual proclivities, or homosexual acts? It's an important nuance.
Judiasm/Christianity/Islam teaches that humans have a proclivity to sin (fallen nature), but God commands them to not sin, i.e. to act ccontrary to their nature.
There is no theological contradiction (or least, not anymore than normal) between natural homosexuals and God forbidding homosexual acts.
EDIT: The actual consequence of natural homosexuality is a difference of understanding. Birth condition vs self-inflicted. "Crack baby" vs "crack head".
Not if you believe in total depravity combined with unconditional election. Which just happens to be the cornerstone dogmas of one very popular Christian denomination.