Per your request to avoid 'derailing analogies', I will try to be as literal as possible.
I'm responding in good faith. I do not intend to be condescending, nor do I intend to lecture you. This is my opinion and it is an opinion I can back up with facts and examples when needed. I am responding to you and I am paying no mind to what other people may have to say about anything included in this response. I do not care what TWS or Novice or anyone else might have to say.
I say all of this because I don't want this genuine answer to be riddled with personal vendettas or disagreements.
If gender is truly unrelated to sex and it is gender, not sex, being transitioned, then surely the physically sex characteristics like hormones and genital appearances wouldn't be of the slightest importance to need to artificially alter to transition gender. Yet... this is how they say is the 'truly freeing' transition, apparently. Why is that so?
Some trans people experience gender dysphoria (distress at the fact one's physical appearance does not align with the gender they identify as). Not all trans people do. Some trans people only experience gender euphoria (happiness at the fact one's physical appearance now aligns with the gender they identify as when it previously did not).
Truthfully, it doesn't matter all that much if a trans person experiences dysphoria, euphoria, both, or neither. I find myself more towards the euphoria side. All that matters is that a trans person has an innate sense of gender identity that does not align with the sex they were assigned at birth.
We have previously discussed at length if transitioning is good for mental health or if a trans person should transition or not (
here). I have already shown you all of the necessary studies to show that conversion therapy doesn't work and transitioning greatly improves mental health outcomes.
Your responses indicated that you personally believe(d) that there isn't any studies that had certain factors/methodology ("I am not saying this to demean you but it is so obvious that barely any source has even mentioned or investigated it."), but that if there were studies that followed such a methodology, it would prove that mental health is worse in those who transition. That's fine. I cannot debate you on evidence that doesn't exist.
If your beliefs are based off of the assumption that studies that haven't been done yet would support your world view if they were done, then we are already at an impasse. We are functionally already at the point of "The science is wrong."
Your question is only about why transitioning to have different sex characteristics is necessary if gender is unrelated to sex. To put it concisely: when you truly with all of your being know that you are something (in this case, a gender [if this is a 'delusion' is another question entirely]) and identify as such, if you look in the mirror and see something that does not align with that identity, that's going to cause some distress for a lot of people.
You can be something without looking like it. You can also be confused/distressed if your appearance does not align with your inner self even though your appearance does not determine your inner self. These are not contradictory statements. If you looked in the mirror and saw a completely different person, you'd probably be pretty distressed to say the least. Now scale that back to the person in the mirror being 'you', but something about it is fundamentally wrong in a way you struggle to describe. If that can cause even one tenth of the distress that looking in the mirror and seeing a totally different person can cause, that more than explains why someone would want to change their appearance to align with their gender identity.
If this is a 'delusion' or whatever you want to call it isn't what I'm trying to answer. You could make the same argument that an anorexic looks in the mirror and sees something fundamentally wrong with what they see as well. I believe (and have previously talked about why I believe) that these are different things, but all I'm doing is answering why.
This is a satisfactory answer to what causes anorexics to feel distress as well, though obviously you don't humour an anorexic when they want to starve themselves to death. Again: it's another question entirely as to why we should humour trans people and not anorexics. It's a question I'm happy to discuss, but I'm not trying to get off track. We've already discussed it before as well.
Is it possible for the gender dysphoric patient to coherently explain why if they are allied to the queer community, they deny that no matter how masculine a woman acts and feels, she is a woman and society is wring to taunt her and bully her and go 'why do you look like a man?' and vice versa with a guy (let's be real this is dine worse to guys who act femme, way worse)??
A cis woman is no less of a woman for having masculine features. A cis man is no less of a man for having feminine features. I'd like to think we can both agree on this.
If a cis woman's appearance aligns with her identity, it doesn't matter if her appearance is 'masculine'. The same can be said for men with feminine appearances. Trans people operate on this same logic.
I know trans women who want to keep their male genitalia or have facial hair or describe themselves as a 'tomboy' or a 'butch' or whatever. Their appearance doesn't have anything to do with their gender. They are changing their appearance not because they feel they feel they must in order to be a woman, but because they want their appearance to align with their identity.
There are cis girls who hate having long hair or wearing traditionally feminine clothing. They might feel distressed if they are forced to have long hair or wear dresses. They are more than able to cut their hair or wear androgynous/masculine clothing if they so please. They aren't doing this because they aren't women. They're just cis women who want their appearance to match their authentic 'self', whatever that may be.
There are trans women who might hate having short hair or being forced to wear masculine clothing. The same can be said of some cis men. Both these cis men and these trans women might be distressed if they're forced to get their hair cut or wear masculine clothing. Both groups are more than free to grow their hair out or wear skirts or whatever if that aligns with their sense of self and desired appearance/identity. A man who chooses to wear a skirt because he thinks it's the best way to express his identity can do so without his identity or gender changing. Trans women are no different.
There are some women who think think they look too androgynous/not feminine enough for their liking.
There are some women who think they look too feminine/not androgynous enough for their liking.
These facts can both exist without saying anything about the identity of either.
People want to look like what they want to look like. For some people, they want to look like the other gender (this doesn't even mean that they identify as the other gender [femboys and tomboys exist]). You can want to look like whatever you want and you are more than welcome to change your appearance to match that desire so long as it doesn't hurt themself or anyone else. This doesn't need to have any relation to gender identity.