This is such a weird take. You're sort of conflating gender assignment with gender perception, while sort of denying gender identity exists, while sort of asserting that gender perception should be authoritative (here lies the bigotry, IMO)

Firstly, the half-denial of gender identity existing / gender applying to individuals

I understand that some people don't identify with the gender that has been assigned to them[...]
Unless your intended tone was mocking and there are some heavy use of air quotes that you did not write, you already accept that a person has their own perception of their gender.

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.
But how others see you is not "gender assignment". How the community sees you is not "gender assignment". It is gender perception.

The purpose of gender assignment is equally for recordkeeping and for practical purposes in early childhood. Gender is typically assigned by the doctor attending to childbirth, and it's done in accordance to biological sex characteristics because as another commenter mentioned, the majority of individuals are cisgender so it's a sensible default. There are some differences in how you might use the bathroom depending on your genitalia, and (for better or for worse) many social norms, like modesty, are imposed/reflected through gender as well. I think parents of a child also have the authority to choose how they present their child's gender to the world, but this is not something to be done trivially or without the child's best interests in mind, e.g. for attention. It's important to note that one's understanding of their own gender identity is something that develops over time, and will develop differently for everyone.

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.
Not really. It is more accurate to say that the importance of the balance of others' vs. our own perception is itself a spectrum that will be different for everyone and can change fluidly over time. The battle between how much to value each is in fact a core component of many trans experiences.

On gender perception:

How you see yourself is only part what makes you, you. You are also defined by how others see you.
Have you ever heard the expression "we are not our thoughts, we are our reaction to our thoughts?" The simplest way to put it is that mere perception is not enough to be a definition. The only authority on someone's gender identity is that person. Gender expression is a cognitive shortcut. Sometimes gendered expressions are intentional, sometimes not. There are also agender people who place little or no value on gendered expression in general. If you're interacting with someone in good faith and you are genuinely unsure how you should refer to them, it takes no time at all to simply ask if they have preferred pronouns. It should go without saying that it's not your business to dispute anyone's gender identity, whether they're strangers or otherwise. You are free to think however you want, the problem is expressing those thoughts in ways that cause harm. Like, just don't be a dick?