Thing you are all missing:
Sex is not gender.
Gender is not sex.
Pretty much every trans* person I know, is well aware that they are 'biologically' defined as male or female (although when was the last time you had your chromosomes tested? It's possible to be XY with a vagina due to genetic weirdness, or look biologicaly female but actually be XY or have extra chromosomes.)
What I believe trans* people are saying is their GENDER, their 'internal' feeling of being male or female, doesn't match their physical sexual organs. Some can live with this, some can't and transistion, having physical changes made.
This isn't helped by the fact that the English words for GENDER and SEX are basically the same, and they've been used interchangeably. So someone can have the internal certainty they are male (i.e. gender), but 'biologically' they present as female (i.e. sex).
Trans* people do not believe they are becoming a woman as far a medicine would need to know, they're not stupid, a transwoman is still aware she is at risk of prostate cancer, and a transman can still have periods, they KNOW they are female gender. This is a subtle but important difference.
That laminated card? It's because they don't want to assume your sex, and therefore are asking in case you have a body part that doesn't actually 'usually' exist in the sex you appear to be.
I'm sterilised. I still get asked if I'm sure I'm not pregnant. They can't assume that I'm not fertile, because that's dangerous.
And, yes, you may utterly KNOW you are female. Think of it it this way, if your brain was taken out of your body and able to keep functioning, the vast majority of people on here would KNOW you were female. Insintricly. Unrelated to where your brain was - on a shelf, in a male body, in a robot, you, yourself, would KNOW you were female. That is your gender identity, unrelated to your sex.
Personally, I don't have that KNOWING feeling, I can't comprehend how someone can know they're male so much that they want their body to reflect that. I don't know I'm female. If you took my brain out, I don't have a gender identity - I'm agender. However, my medical sex is female, so I still have screenings, and check my breasts, and I've had kids. But I've spoken to others who do have that KNOWING feeling, trans and cis, so I know it exists. And just because you KNOW your gender is female and your sex is female, doesn't mean this matches up for others.
And pronouns? Welcome to language. Is it going to hurt you to call someone the pronouns they want? Can you remember someone's name? And their preferred drink in the pub? You can probably squeeze in if they use 'he', 'she' or 'they'... It's just another preferrence, it makes them feel happy, and it's not actually hard once you get used to it. Personally? I use she/her as it's easier for everyone as I have a very obvious big chest, but I couldn't give a damn if you use they/her/his with me. Other people will care.
Also, 'exclusive female health issues'? Transmen can have periods. Their gender is male. They're entitled to a view as well.