Yes, and they're also following the playbook of their hero Viktor Orban, just reelected in Hungary. He's banned the teaching of gender studies in Hungary, censored and limited images and stories of/by/about LGBT people in entertainment and media, and as a Hungarian LGBT activist writes:
"the government's election campaign message is that Hungarian citizens and their children are threatened by 'homosexual propaganda' coming from the 'decaying West.' And that Orban, once again, is the great protector of traditional national values.
Being a scholar of gender studies, I could lecture endlessly about how this government's approach to social issues is patriarchal, sexist, homophobic, transphobic, nationalistic, racist and xenophobic. Or at least, I might have, until my very discipline was banned in Hungary in 2018.
Being a lesbian activist as well, this rhetoric has had an emotionally and psychologically devastating impact on LGBTQ people in Hungary. Our mere existence is called 'propaganda.' We are treated as an enemy (of whom exactly, we might ask), our sexual orientation and gender identity is conflated with pedophilia so, by a simple association, we are criminals.
In this discourse, children are treated as blank sheets -- currently asexual but future heterosexuals -- whose 'normal' sexual development may get 'corrupted' by adults. According to this world view, if you read a fairytale about two princes who fall in love to your child, you are guilty of turning the otherwise 'normally developing' child into a homosexual or transsexual. Because of course, no children would become an LGBT+ person otherwise."
So basically, LGBT people are just pedophiles who groom otherwise straight "normal" kids to be gay and lesbian and transgender. And if you can eliminate that influence, then almost no one will become LGBT as adults. It's completely false of course, but that's already the Orban line in Hungary, and Florida's "Don't Say Gay" law is the first step in that direction here, and will no doubt be copied and expanded upon by other GOP-controlled red states.