I understand where you’re coming from and, if that works for you, great (it works for me too).
I also think, we live in a society where labels have power and meaning. Being able to identify with stuff like girl power, because you are a girl or no shave november if you’re a man can be a part of your identity and help place yourself in our society. Of course, you could try to participate in these initiatives without belonging to that label (or gender) but, for many people, the label makes it easier.
In the case of transgender, I think many transgender people make it a point of saying they are of their chosen gender. The label becomes relevant however, when we’re talking about trans rights or discrimination, or when you are looking for people who share your challenges as a transgender. I’m ignoring here, the fact that transphobes assign a lot of importance to the “transsexual” label, they tend to disagree with your premise that gender is a social construct.
Good one!