“How have the tropes of freedom, democracy, and gender equality facilitated the current Euro-American ambition to remake Muslims and Islam?”
A couple of things come to mind from this question: Missionary feminism uplifts western ideals of womanhood and dismisses Islamic practices as oppressive. This question also brings me back to the conflicts between radical libertarian feminism and radical cultural feminism. In both, there are defined rights and wrongs in being women. To engage with sex work is offensive in radical cultural but accepted in libertarian. Natural reproduction is not encouraged in Libertarian but is applauded by Cultural. The divisions between people purely based on choice, on how to live is particularly relevant to this question.
Freedom is subjective. Gender equality and how it looks societally is subjective. Constraining either term in the context of one world view goes against the promise of freedom and equality.
So, how have the tropes of freedom, democracy, and gender equality facilitated the current Euro-American ambition to remake Muslims and Islam? The ways in which these tropes are portrayed in media and society influence the outlooks of people living in the world. If you are always told and exposed to one ideal of gender–thats how gender equality and freedom look to you.