It's not about your "existence" any more than political debate about immigration is about my "existence." You exist, I exist. But what obligations do other people have to us? What accommodations should society and individuals make, and which do they have no need to make? What aspects of culture should change or shouldn't change? What should we teach kids in school about complex subjects? These are proper subjects of social and political debate.
Recall that this is happening against a backdrop where 90% of Americans, including 80% of Republicans, support employment protections for LGBT people. Bostock is already the law of the land. So the obligations that are in dispute here aren't even basic civil rights, but instead broader social and cultural changes. People have every right to vigorously debate proposed changes in the social and cultural norms they're expected to follow.