You do not have to “prove” your gender by forcing yourself into societal roles that may not fit. You are not “failing” anyone by fitting into societal roles that are comfortable. It is not your job to break down the binary/patriarchy/or anything else. If you want to, go for it, but you have no obligation to do anything for cis people just because you are trans.
People don’t realize that’s what I was doing for two decades: trying to prove that I belonged as a woman, and that I was even a good one at that. But I was really trying to prove that to myself, and you know what? Can’t prove something is when it isn’t. Now I still have the urge to prove myself as genderqueer, but I’m really past that. Either who I am exists in the minds of those around me, or it doesn’t. I’m here, I’m open, but I’m not responsible for their lack of education.