Pretty good run down. I know i'm glad I transitioned in Nevada, where I had protections in case my landlord would try to kick me out (a very real concern of ours.)
Yep, there are differences between cis and trans women, but we should both be treated as women. I've never said anything like "I am like a cis woman in every way possible! Even though I have XY chromosomes and lack a uterus and ovaries!", I'm saying "I have enough in common with cis women that I should be considered the same, and my genetics and lack of reproductive capacity shouldn't be a factor."
Sex is ultimately irrelevant in this discussion, but they keep trying to bring it up in order to muddle things. It's essentially genetic determinism, which is a concept that has some... unfortunate consequences throughout history. This is all about gender, which is much more complex of an issue.
And I have never had any desire to get rid of single-sex areas. As a woman I have come to realize just how important those spaces are to us, they provide a measure of safety and security when we feel we need it. It is just another strawman.
And imo, I would just ignore the people trying to call you a transphobe, they are just projecting their own ignorance and bigotry onto you. They've tried to do this to others like LondonJohn with no success. "Trans women are women." never meant there are no differences. Just that we are treated no differently unless absolutely necessary.