There's no such thing as gender separate from sex. There's the recognition of one's immutable, inherent, sex, and tacking social expectations on top of it, but never that one could choose, or "feel". Always derived, never a choice. And when people allowed cross-dressing, it was always clear it was fake, pretending, never true. But they allowed people to have their personal delusions.
The origin of this use of "gender" itself is due to the prudishness of English upper classes in pronouncing the word "sex", so they repurposed "gender" which is just the French word "genre" meaning "kind" or "category". Much more acceptable in polite company than something that can allude to a sexual act, fornication.
You seem to be partly arguing from a position of ignorance.
The trans-ness some people experience is extremely general and durable, far more consistent with the explanation that they innately are their gender somehow[^1], than with choice or psychosis. Some people feel pressured by this to, despite all the societal dis-incentives, medically transition. They then are not only their gender in behavior and reported experience, but also physically (with the exception of some hard-to-change stuff such as fertility).
We usually handle such general, durable "personal delusions" by accepting them. If I studied some math for years, can do said math and am employed at my local university doing mathematics, I am a mathematician. I do not have delusions of being a mathematician. If I move to, say Germany, and after years speak the language, have children there, participate in the local culture, and have a citicenship I am now German. Only the most backward people would say I have delusions of being German. Although, this cultural rigidity of course exists, I do not see it as desirable. An advanced society should accept and accomodate its outliers instead of steamrolling over them and making them suffer.
[^1]: Afaik currently a neuroscientific explanation is not forthcoming
The "tacking social expectations on top" is the part that is gender!
There's no biological foundation for wearing a sari, hijab, miniskirt, etc. Those are social expectations for women, or part of the role women fill in society.
It's a wholly different concept than biological sex. My penis does not make it impossible to wear eyeliner. But society has a social expectation that I do not. It's not a sex characteristic, it's a gender characteristic.
You might believe gender is immutable. I'm not going to argue that with you. But denying the idea that humans have both characteristics derived from biology (sex) and from societal expectations (gender) is simply objectively incorrect.