Since we're talking about trans people I might as well share my own take:
I've for the longest time been a feminist, in the sense that I think discrimination based on gender is wrong. This argument after a while morphed into the argument that one shouldn't really consider gender to be important or ideally even acknowledge it since it often leads to sexism.
The existence of trans people spit in the face of that argument, since they are people who loudly proclaim that "Yes, gender IS important, otherwise I wouldn't want to change my biology.". That rubbed me the wrong way.
After giving it some thought I reached the conclusion that it was a bunch of baloney, since sex has a clear biological definition, and attributing sociological aspects to it would by necessity define sociological attributes to gender which would constitute sexism.
I then went about my day, satisfied with having reached a conclusion, until I suddenly stopped in my tracks, turned my head around, a sudden thought having entered my mind:
"Do I know any other words with a clear biological definition which also have a sociological definition which is not by necessity the same?"
I realized that Yes, I do: "mum" and "dad". That has a clear biological definition, but the fact that people get adopted means that the meaning is not always the same.
I then realized that when I called it baloney I did the equivalent of suggesting that a relationship between a parent and a child is lesser due to a lack of shared DNA, which made me feel bad, and reject my earlier assertion. I furthermore asked myself if my relationship with my parents would change if they suddenly called me up and told me that I am actually adopted. After some thought I realized that "No, nothing would change, I love my parents", which made me realize that in everyday speech it is the sociological definition we use independently of the biological one. Same thing with gender.
I've also found that not even bringing up a sexist subject for a fear of reinforcing it often works as a way from shielding people from hearing about sexist occurrences, which prevents progress.