The nice thing is that if a 14-year-old wants to be a prostitute or to get a swastika tattoo, nothing is lost making her wait a few years to decide when she reaches adulthood. She can do it then, having lost nothing but time.... and I'm unaware of any evidence that not being a prostitute or not getting a tattoo causes people severe emotional trauma in a way that forcing them to live for years in a body that conflicts with their internal sense of gender is said to do.
What makes medical transition of minors a thornier issue is puberty. The longer you wait for such treatment, the less effective the treatment will be, in the sense that the body will have undergone massive and irreversible physical changes by the time the person is an adult. Plus, as I mentioned, the claim is that merely being forced to live in that conflicting body for years can cause such emotional trauma that suicide or other self harm is greatly increased.
So, it isn't a question with a simple solution, the way those other two examples are. Merely postponing the decision doesn't leave the person in more or less the same place a few years later, when she can decide as an adult. A decision to postpone transforming the body in one direction medically is a decision to transform the body in another direction by way of normal hormonal processes, with potentially permanent psychological harm resulting.
Perhaps that's better. Perhaps not. That's the evidence I'm hoping to see to take a position.
Anyway, a better analogy would be whether a 14-year-old should be allowed to get an abortion. That has that same "ticking bomb" set-up for a transgender person dealing with the changes that will come with puberty, along with the same extremely high psychological stakes. You can't really just postpone the abortion decision until the person is an adult, because postponement is effectively a choice to force huge long-term physical and psychological consequences on the person.
You have described the reason why chemical castration drugs (such as lupron) are so loved by transgender advocates, and so feared by many others. Without the drugs, children will start looking like the sex they do not want to be. A girlish boy starts looking (and talking) like a man, and a boyish girl starts looking like a woman. A child might genuinely and deeply want to become the opposite sex. Or maybe the child just doesn't want to become an adult. Or maybe the child is going through a phase. Or maybe the child is just extremely confused.
What does it really mean to be a man or a woman? No one can actually say what it means. Most women have no interest in big military machines, but lots of men do. Most men don't care as much as women about interior decorating or growing pretty flowers. But our society no longer forbids or strongly discourages people from behaving like the opposite gender.
For example, homosexual men often fall into either a masculine or feminine type of personality. Male homosexuals can become hair stylists, fashion designers, etc., and hardly anyone really minds. Or they can live like regular men, and you can hardly tell they are homosexuals.
How much of gender personality is learned and cultural, and how much is influenced by sex hormones and DNA? Well no one knows. I think our medical industry has taken a very wrong turn in deciding that drugs and surgery are the answer. Not really surprising, since our medical industry thinks drugs or surgery are the answer to all kinds of problems.
I think we need to question our medical industry, on this and many other topics. I have distrusted the overuse of certain medical interventions for a very long time. And THAT IS WHY I disagree with transgender "affirming care." It is not "affirming care." It is applying harmful substances and surgical procedures where they should not be applied.
I don't care if men wear dresses and makeup, or if women join the military. I DO NOT CARE. I have always felt kind of gender ambivalent myself, and rejected a couple of my early career attempts because they were 90% female. I never wore makeup or high heels. I do wear dresses when going out, and I have long hair, just because that's what I like. The women I know are all over the place when it comes to this -- some always wear pants and never wear makeup, some wear pants but do wear makeup, some wear jewelry but no makeup, etc.
I really hope our society will take a step back and get perspective, and reevaluate what the transgender activists have been claiming. Puberty blocking (chemical castration) drugs are controversial and we have to question the wisdom of letting young children take them.