However I've always thought arguments about 'being born that way' kind of betray an acceptance that there is something inherently wrong with the sexuality of the person being judged - you don't need the excuse of it being 'not your fault' to be allowed to have your sexuality, whatever it is.
I agree.
I have thought this for years, but I rarely say it because IME people are very invested in the "born that way" story.
"You can't discriminate against them, because they were born that way."
So if they had chosen it, then it would be OK to deny them equal rights? 
I think that's why the bisexuality fears - they could "choose" to be straight. Their homosexuality is not an undeniable biological fact, an unfortunate situation thrust upon them by nature. It's something they do as it pleases them - hence greedy, untrustworthy, phase, etc.
It makes so little sense to me that sexuality, which is so complex and not static, but involves so much learning and habit, could be pre-programmed by birth.
Which is not to say we actively choose who we are attracted to, obviously. But I think it is, in part, shaped by our choices and experiences and the people we know and the culture we grew up in.