You only know your own gender. Society refuses to accept the existence of nonbinary genders, but that does not mean they do not exist. This is like saying, "I like chocolate ice cream but not vanilla, and I have heard that many other people do like it, so everyone likes either chocolate or vanilla!"
A range of masculinity and femininity exists; No one is the same. In a spectrum it's impossible to define each point. A spectrum of gender has been accepted since the beginning of time, and the archetypes of beauty are not at the extremes of that spectrum. So we've always celebrated "non-binary".
I get the point you are trying to make, but I think you have gender and gender expression mixed up. You can express yourself however you want. Anyone can be very masculine, feminine, or right in the middle, regardless their gender. You can like vanilla, chocolate, or any other flavor.
"Liking" is an analogy. I am not saying that ice cream corresponds to gender; I am saying that liking ice cream corresponds to having a gender. I understand what you mean, but if society (personified) asks another person what their gender is on paper, they will leave two boxes, not a blank space.
The amount of people who identify as neither male or female is so small that it can be discounted as an anomaly. There is no utility in building anything on anomalies. It's good enough to say "other", for rare cases, and leave them alone. Truth is never absolute. Being 99% correct is acceptable.
There are only two genders, and gender is not a social construct. Women have XX chromosomes and men have XY chromosomes--this can be seen even before birth. You can't possibly be another other gender because gender is not a feeling; it's biological.
Intersex people with XXY chromosomes are just males with an extra X chromosome—still male. Being born intersex is very rare anwyas, so why are you bringing this up? Not everyone who claims to be ‘non-binary’ is intersex. Intersex itself is not a separate gender.
Gender dysphoria is completely binary. Gender dysphoria is the sole reason a trans person transitions; it aliviates their dysphoria. When a trans person transitions, they can only transition to female or male. How can you transition from being male or female to being nothing at all?
Gender and sex are very closely related, and even the same thing on some legal documentations. And, if gender is mental, explain why people with gender dysphoria feel the need to change their physical appearance.
Your brain is responsible for the mental; It is a physical organ. Therefore gender is physical. Brain structure, and chemicals are of great importance in determining your behaviour. You could say a woman might be born with a more masculine brain, but that only proves that gender is physical.