They're fighting the Republicans attempts at blocks. I'm not sure what you mean by "sitting back and letting Republicans control the narrative" Biden and Harris are already vocally condemning the transphobic words and putting out their own counter-media. Could they do more on top of material change?

And, from where I'm sitting, it looks like they know with the narrative, they have to fight an uphill battle against a "flood them out with attention grabbing outrageousness" strategy, they seem to have put more focus on the material change aspects.