There's a lot of conservatives that don't care one way or the other. The majority of conservatives I have met in my life really could not care less about homosexuality, meaning oppressing them as well. It's just a popular image to portray conservatives as "all gays are sexual deviants meant for hell," if the conservative in question is even Christian. They might just like guns and dislike immigration. 

Of course there is a larger population of conservatives that think that way compared to liberals, but by the 'wrong century' that catchamber meant, is that in the 19th or mid 20th century no one questioned that gays were inherently bad (at least I think thats what he meant). 

Also my parents are liberal democrats and still think Trans people shouldn't be allowed in the bathrooms they identify with so.