Transgenderism isn't a word, and its use tells me that you've not spoken with many trans people about the words we do use to describe our experiences.
The way you're using it takes a descriptive adjective and turns it into a condition, which many trans people (myself included) find dehumanizing. Even though your piece is fairly neutral, the use of this non-word made it hard to digest. It made it abundantly clear that you are talking about us to cis people, not talking with trans people about our triumphs.
It's furthermore worth noting that "transgenderism" is most commonly used by transphobes and gender-critical types largely because so many trans people have stated that we don't condone or appreciate its use.
"Transgenderism" is a word that effectively excludes us from the conversation, because it fails to adapt to the words we do use to describe ourselves and our communities. I personally would recommend avoiding this word. I do not feel that you are using it in malice or as a way to exclude trans people from the discussion, but that is what its use will do.
A good way to rewrite your title, for instance, would be, "Trans Rights Won the Culture War." Most everywhere else you say "transgenderism" could also be replaced with "trans rights." This reframes the discussion from describing transness as a condition and moves it toward what we are, in fact, fighting for: the right to exist as we are.