I have similar thoughts for a solution- just don't have any sex or gender indicators on ID cards. Have a whole separate medical card that isn't for identification. You keep it in your wallet in case you're ever in an emergency, and doctors will use it to know if they should treat you like an xx or an xy or something else if it so applies. It could have a list on the back where you write any current medications, so a Dr knows what interactions your body might have to any treatments. This would be useful if, for example, you're a female presenting biologically xy Trans person who shows up unconscious in an ambulance. Then the doctor can look at that person's card and say "I will give this person the medical care that an XY person needs" vs saying "this person's ID card says male but they look female, should the treatment be for a man or a woman?"

As a non medical professional I'm sure I am oversimplifying the ordeal and glossing over important points, but it's just something I've thought a lot.