Keep in mind, this is not necessarily what is consciously going through people's minds when they think about gender identity. Cis women certainly don't have to think "I am happy with the way society treats me" to identify as a woman. They don't even have to feel "comfortable" with the way they are treated as a woman. It's hard to separate the "experience of being a woman" from "being a woman" when we are the sum of our experiences.

But that is also relevant to what it means to be trans, because it's not just about identifying as something else, it's also about not identifying as something, too. And again having to navigate a world where most other people are using gender as a proxy for sex and sex as a way to determine part of how they treat you means that there are advantages to having an outward expression of gender that fits with the expectations of others.