"gender is a social construct, it's just a part of a person's identity, allowing it to be what we want"
You misunderstand what identity is. Your identity is not what you feel about yourself or how you think about yourself inside your own head. Look at your identity card. It lists the important details about you that OTHER people use to distinguish you from anyone else: your date of birth, your place of birth, your sex, your name. Inside your head you don't need a name, you're the only person in there. Your date of birth is irrelevant inside your head. Your identity is what other people agree on to say that you are this person and not that person. As such your identity absolutely doesn't allow you to be what you want, it is not your decision to make.
"People choosing their gender doesn't affect your life, so why do you care?"
I don't care how you present. I do care if you claim to actually be a woman, because that is objectively false, and misogynistic, and endangers women.
"Sex (or genitals) is indeed a biological reality, but its only importance is for when you want to have kids, else we don't give a damn about what's in people's pants."
There are lots of people who don't want to have kids but very much do care what is in the pants of people they want to have sex with or form emotional relationships with. The fact that you might not, should not be used to invalidate those people.