Yes and no.
I suppose it’s like socks — for the most part, we put them on and forget about them, and everything is fine, until it isn’t. If we get a pebble caught in our sock, it rubs and irritates us until it’s hard to walk – hard to concentrate. In the same way, I think people tend to wear their gender without thinking about it, and everything is fine — until it isn’t. When the gender we put on doesn’t fit us correctly, it is irritating and painful, and can eventually dominate our thinking. This is what leads some people to feel that transgender people are obsessed with gender; we often do spend a lot of time thinking about gender, but it’s more because there’s something wrong and we need to address it.
As transgender people address their gender and begin to make the changes needed to better align their experienced gender with what they present and express, the mismatch gets smaller and less irritating, and so gender becomes less dominant in our thoughts.
I am a few years into my transition. The first year, everything was about gender for me, because everything hurt and the mismatch between my gender and the one assigned to me was a freaking boulder in my shoe. Now, I’m pretty far along in my transition – I’m living as myself all the time, have finished my legal and social transition, and have made significant progress on my medical transition. Gender no longer dominates my thoughts. I am (finally) living authentically as the woman I’ve always supposed to have been.
But interestingly, I’ve found that for people who struggle with accepting me as a woman, their thoughts have been increasingly dominated by gender — specifically, how I don’t conform to their expectations for how I should perform gender to what they want me to be (not what I actually am).
So yes, gender is important, and that becomes more evident when there’s a problem with your gender. And no, it isn’t important, once that problem has been resolved.
Just like a pebble caught in one of your socks.