First, I’d challenge the ‘most people’ part. In my experience most people really, truly do not care one way or the other if someone is trans. But to be fair, this may vary between countries, societies, etc.

Next, aside from its obvious use as a political wedge issue, and generic fear of the unfamilar, I’d argue that a decent portion of conservatives (in a North American sense) fancy themselves as Christians. There are ideas in the bible re: the ‘place’ of men and women in relation to one another that make deviations from this (gender identity not match genitals, homosexuality, etc.) aborrhent. I personally think those people are missing the actual good things to take away from Christianity, but I’m also not a Christian, so idk.

I think it’s silly, really - coming out and living as a Trans person, if that’s something folks feel they want/need to do, is one of the most radical exercises of a person’s autonomy and personal freedom. I’m for it.

Complete aside, but it’s always weird to have conversations with folks who like the idea of transhumanism who are transphobes. Met more than one conservative person who is like this, which is really the only time I give trans people much serious thought beyond “Oh look, a fellow human. Neat!”