Right-wing culture war "issues" come from the top down, they are manufactured, fake problems to distract the people from relentless capitalist exploitation. "The reason you're so angry all the time is not because the rich are destroying everything that makes your life livable, it's because they let men into the lady's room! The world has gone mad!"

Fox News and Republican candidates talk about them because they're cooperatively speaking them into existence, appealing to an audience they themselves create in the process. But at the end of the day, this audience is quite ephemeral because culture war "issues" simply don't have a real basis in reality. Transphobia, bathroom bills, anti-abortion, all of these topics are not actually that popular with the average Republican voter.

The number of freaks who seriously care about culture war nonsense is way smaller than media discourse would have you believe. Right-wingers have to talk about fake solutions to fake problems because every real solution to real problems would involve getting rid of them. They don't talk about Disney transing the youth because it's what their voters care about, they talk about it because if they talked too much about the real problems, they'd risk people figuring out what's actually causing them.

Edit: Republicans aren't winning because Americans hate trans people, Americans hate trans people because Republicans are winning. Whatever culture war issue Fox News is babbling about is completely interchangeable with any other one and their voters will stop caring the second they stop talking about it.