Which political party has made Transgenders a core issue of the party? It’s the Republican Party.
That bozo in Florida made fighting the “Woke” a cornerstone of his campaign and even the Republicans sent him home packing before the second primary.
You are right that majority of voters don’t care one way or the other about the transgender issue because most people will go through their entire lives without ever interacting with a trans person. The most well known progressives (Bernie, AOC, Liz Warren etc) don’t make woke ideology a core part of their focus. They spend the vast amount of their time pushing policies that help people in real life like Medicare for all, living wage, income inequality, climate change etc. republicans are the ones who make fighting silly cultural issues a core part of their party.