They are fighting for that right because they are intersex or trans

Where did you get that? The article indicates that they are non-binary. That is neither intersex nor trans.

That’s the crux of the issue here: someone does feel the need to not fill out the field (i.e. filling it in with an “X”), and the health authority is forcing them to use either “M” or “F”

But the person in question is non-binary, which is a gender identity unrelated to their sex. It is perfectly possible that this person feels perfectly okay with a health card that indicates that their gender identity is non-binary while their sex is binary. In that case, separating gender from biological sex in their health card would address the issue at hand, particularly since nothing would prevent both fields to be left empty if they so choose.

As for why it makes sense to specify the biological sex in their health card, it is a medically useful piece of information that 99% of the patients would have no problem recording. Don't forget that patients are not always able to communicate at the time they are in need to health care. So, again, if it is useful and non-controversial for 99% of the patients there's no reason to remove it, just provide a way for patients to opt out of a simple M/F choice if they wish to.