Perhaps I should try and expand on it a bit then, although I'm trying to articulate this for the first time so bear with me if I end up talking shit.
I think most people have an inherent understanding of what sex they are, which they have understod since infancy and they are largely at peace with. They may not like aspects of their body, or even aspects of their sex such as menstruation, and they certainly might not like the gendered assumptions that come attached to their sex, but broadly they don't seek to change anything, don't hate their genitals, can function adequately sexually within their sex, and don't deeply desire, or feel they should have the body of the opposite sex. For the old school mumsnet approved transsexuals however that's not the case. The classic 'transsexual' child often doesn't just claim to want to be the other sex, but insists they are the other sex, often from as early as they can articulate it, and they keep insisting it until they become gender critical shills or die. I think it's probably more accurate to describe this as sex dysphoria rather than gender dysphoria - they are at odds with their bodies, not their social role, and this often led a lot of people to look for neurological or biological explanations for this that are rightly read as regressive - having the wrong body for your brain etc. I think people sought to medicalise their condition as away of deflecting from the shame they felt at being that way - it's not my fault, I was born this way - and a way to seek social approval for the same reason the emerging LGB movement sometimes made similar claims.
Under this model, and due to the hostile social conditions of the time, it's likely only the most sex dysphoric would go as far as transition - I mentioned upthread the first two trans women I ever met who told me not to transition unless I was suicidal and felt I really couldn't live another day as a man. But this model didn't really work for a lot of people, and felt like not just surrender to transphobia but also a form of gatekeeping from the older 'true trans' population. Sex dysphoria can be of differering strengths, and manifest in different ways and this was always the case, and what happened is people who didn't feel compelled to physically transition (or were too scared to) found solace in the LGB movement in things like the drag queen or butch communities, or if born physically male ended up in the transvestite quagmire - a highly sexualised identity which largely developed in the interests of male fetishists and those males who wanted to fuck them and alienated everyone else. But there were always transvestites who were not sexually motivated, who lived in the opposite gender to their birth sex all or most of the time (when they weren't at work for example), but who didn't seek surgery. Also there were always people who felt somewhat in the middle and were somewhat adrift although people were calling themselves genderqueer as early as the 90s if not before. And when a new generation emerged and started to build a language and way of articulating differing experiences of both sex and gender this was ferociously resisted by a lot of older trans people, who saw it as a threat because they didn't want to be visible or politicised and felt it undermined their own struggles. So there was an almighty row in the trans community abut it, which ultimately got resolved by an understanding that everyone experiences these things differently, they can be difficult to articulate, and probably the best thing to do is respect people's identities and not dictate to people what they should call themelves or what counts as being a legitimate transsexual. But I guess this is where the relationship between sex identity and gender identity started to get even more complex and influenced by things like aesthetics, preferences, and even political statements.
But even without this sex, and gender, and the interplay between them is still complex. Presentation doesn't just end at clothes and make up for example, many cis people alter their physicality to more closely match their assigned gender whether that's shaving their legs, getting ripped at the gym, having breast enlargement surgery etc. The pressure to conform to the gender assigned based on our physical sex is immense, and our internal maps of ourselves are unlikely to be able to truly untangle the two - it's been imprinted on us from birth, and those whose sex identity is askew may look to those of the sex they seek to be and emulate them. So someone with less intense sex dysphoria, or a mismatched sex identity, or a blurry sex identity, might gain comfort from changing their gender presentation and social role and this is enough for them - without a need for surgery, because that's actually quite a big deal, comes with consequences and is prohibitively expensive or very difficult to access in many countries (including the UK). But what underlies it is discomfort with their sex,and I suspect if you said to most binary trans people you can change your physical sex by waving a magic wand they would jump at the chance because it is bodies that lie at the core of the discomfort not stereotypes or social roles. In fact I suspect if you said to binary trans people you have a choice between remaining in the sexed body you were born with but changing your gender role, or changing the sexed body you were born with but remaining in the gender role you were assigned based on your birth sex then most would choose the second as the least worst option. When those who go onto transition, or seek trans healthcare, talk about gender dysphoria they often really mean bodily dysphoria based on their physical sex, and this may be present and a motivating factor to a greater or lesser degree in lots of people whose gender identity is not in accordance with their physical sex.