Anti gay/lesbian is fading quickly from most conservatives as the older ones die off. Most young conservatives don't view gays/lesbians as an issue.
In the UK, it was the Tories (conservatives) who gave us total equality in marriage and related laws. Labour (liberals) only gave us business contract based "civil partnerships" and "somewhat similar" rights to marriage.
Trans and identity issues are the newest thing. NYC, Canada and maybe others legally punishing people for "wrong speak" when it comes to personal pronouns and crap like that. In NYC, a business can fined $125,000 for ACCIDENTALLY using the wrong personal pronouns and/or mis-gendering someone; each time. It's $250,000 each time if it's deemed to be on purpose. WTF?! Imagine girly girl Elliot Cope walking into a shop in NYC and someone greets her a if she weren't male. Everything about her is female, but she claims to be male. She is a female, she presents as a female, but claims she's male. She could kill a business with fines just because she's an attention seeker.
Real trans-sexuals aren't the problem/issue. It's the gender trenders, gender terrorists and all the other ass clowns.