A lot of the right-wing discourse seems to be populistically pandering to cultural preferences among their constituency, but for something like this to really catch on as much as it seems to have, there needs to be some fundamental, almost primordial instinct around which the discourse revolves, whether thats in good or bad faith.
In this case, it seems to be about protecting children, and here Padraig has given a few examples of conservative-leaning think tanks and social welfare organizations which have confessed to exploiting this notion of "protecting children" to curb trans/queer-related cultural and political developments they find disagreeable by conservative standards.
I suspect a, if not the, primary thesis driving this conservative discourse, however explicit or implicit it may be in different instances of this sort of messaging, is that your kid may be doing "fine" until they're exposed to gender-fluid/non-binary discourse, and from that moment a gender/sexuality-related complex is seeded in your child's mind.
Personally, I suspect the risk of this is negligible, although I'm sure there are some cases out there - if we can entertain the notion of being able to compare alternate realities and trace causal provenance of events as they fork into these different realities - wherein a child isn't experiencing any gender/sexuality-related identity dissonance of any kind, but are on the cusp of adolescence and thus about to have their sense of identity largely thrown into flux, and when they encounter discourse relating to gender fluidity, they subconsciously attribute their adolescence-related identity turbulence to some gender/sexuality-related dissonance they didn't know they had.
Anyway, that was my best attempt at articulating a viewpoint which I don't ultimately agree with, and which, while I understand that particular framing of it, I think has resulted in undue harm to a number of people experiencing dramatic difficulty squaring their identity with the world around them.