In the US, it was the left who decided to push gender identity into law and policy, with no regard to the adverse consequences of doing so. That the right decided to capitalize on this for political reasons is just them taking an opportunity that was basically handed to them on a plate.

Interestingly it's a bit different in the UK. Both the main left and right parties had been promoting gender identity based policy for years, and it was only though the dedicated efforts of feminists who pointed out all the problems with this, and particularly the negative impacts on women and girls, that it recently started to be reversed.