I don't think they care all that much about trans people as such, it's the idea that anyone can escape the rigid social rules they have in mind.
This is merely the first step in their attacks on any sort of deviationism. Trans people are simply an easy target. Once they attain this, they go after gay and lesbian people; in fact, they've already started. The idea is to get a momentum going. The anti-abortion thing is the same idea, go after one form of freedom for women, and then go after others (e.g., birth control), until they end up taking away all of the freedoms/rights that women have fought for over the past 100 years.
Jews, muslims, and atheists, and ultimately anyone who doesn't slavishly adhere to their brand of christianity will be forcibly converted or exterminated.
Look at the fundamentalist christian cults, where women are slaves of their fathers and husbands, and even the men are subservient to the priests. And any thinking that deviates from the dogma dictated by those in power is to be ruthlessly crushed (cf. the Inquisition.) That's where they want to go.
