This is the part I have a problem with:

Gender is a category assigned by the individual

Gender isn't assigned by the individual. Gender is assigned by parents, doctors, the community, and society broadly. Gender is an inherently social construction. Some people have misconstrued this to mean that gender is an individual construction. It is not. How you see yourself is only part what makes you, you. You are also defined by how others see you. I understand that some people don't identify with the gender that has been assigned to them, and thus want to change their gender assignment, and I empathize with those people, but whether or not they are able to do this is at least partially out of their control. Just because you see yourself a certain way, doesn't mean others will see you that way, and, again we are defined not only by how we perceive ourselves, but how others perceive us.

You're thinking in the right direction, but you're not quite there. Yes, gender gets assigned at birth as your biological phenotype (because the overwhelming majority of humans identify as cis gender). Humans become individuals and eventually realize that maybe their assigned gender does not fit them. At this point it becomes their own choice and it overrides any gender assignment given at birth. Depending on the individual, gender changes from something that's assigned to something you assign yourself.

Yes, genders are tentatively assigned at birth because most languages are based on gendered pronouns and gendered words. So we need a gender to refer to this new person, and we can't ask babies for their input.

But that's not different than a parent saying his baby will like sports. Or, in my case, saying your baby will grow up to play Magic The Gathering with you.

Because when they're old enough to decide for themselves, they can change whatever temporary labels you attached to them. And they can say they don't like Magic. Gender. I meant gender.

most languages are based on gendered pronouns and gendered words.

Relevant Tom Scott video: https://youtu.be/46ehrFk-gLk

First, only about 25% of languages are fully based around everything being gendered.

Second English has some specifically gendered words as remnants from old English and the languages that blended together to make English.

Third, of those gendered languages, they don't necessarily agree on what gender things are. In the video, they mention that "A Key" in German is masculine and in Spanish is feminine.