We may not be fully on the same page here. The origin of same-sex attraction is not important. Behavior is what matters and it seems to me that anyone from either of your groups could choose abherrent behavior.
When an individual "immediately rejects your efforts to reach out as soon as they realize you don't condone their sin", it is probably wisest to not lead with biblical verses.
True, but it is a moral issue. If you chose not to appeal to Scripture, you would have to appeal to something, natural law maybe. It can be done.
It also occurs to me that this happens most of the time when talking to liberals - as soon as they realize you don't agree with them, they cover their ears and run screaming from the room.
The origin is indeed the core issue.
IMO, behavior is their concern. The only time it becomes our concern is when that behavior becomes public or they insist we treat them as of the opposite sex; or they insist we spend our tax dollars via the schools to promote our acceptance (there are likely more, but those jump out).
Again, the one eye blue and one eye green scenario, biological mistakes happen.
The born homosexual looks in a mirror and states "that is a mistake". Imagine society trying to convince that individual they actually are of the opposite sex. The success of the argument will, IMO, depend on significantly more than a verbal discussion.
Finally, as I noted upthread, I am confident a large segment of the "born" group, if not all, would seriously consider taking a "magic pill" (or any other demonstrably successful therapy), if they were reasonably certain it would resolve their dilemma.