You don’t have to read tea leaves: https://www.kff.org/other/press-release/poll-large-majoritie... (“In each case, large majorities across partisan lines, including more than 7 in 10 Republicans, think such discrimination against LGBTQ people should be illegal.”). These folks are upset about the “existence of transgender people,” but also want to make it illegal to discriminate against them?

You can understand the “culture war” by applying your own description of the “liberal interpretation.” As you aptly summarized, liberals don’t care about biological determination because they believe in people being free to express themselves without being limited to traditional norms and categories. It was when liberals started to advance that view—for example by promoting drag shows, which are about cis men dressing up as women as a form of self expression—that they triggered the culture war.