I’m not sure you fully understand the issue. This isn’t merely about catering to a patient’s preference to be addressed in a certain way

...which is why I suggested differentiating between the way the patient would like to be addressed from their biological sex, and storing both in the health card.

What is the process for assigning gender at birth anyway? People look at the baby’s genitals and make a guess. For 99% of babies, that heuristic works. But it turns out that some people are trans or intersex, and that wrong guess causes nothing but trouble.

You appear to be conflating gender an sex. Sex is biological, while gender is a social construct. We assign sex at birth, and infer gender from that sex.

In trans people, the two do not match, which sometimes leads to disphoria. The sex assigned at birth is still accurate, but the gender that was inferred from it is not.

In intersex people, their biological sex is difficult to determine and often doesn't fit a simple binary (e.g. XXY chromosomes or androgen insensitivity).

The same solution is to trust that people are whatever gender they say they are

There is nothing to trust about their gender, we can respect their self-identity or we don't. As for their biological sex, it's a significant piece of medical information. If somebody feels uncomfortable stating it clearly on their health card, they can choose not to fill it out, but for the immense majority of us it is a non-issue.