Labeling something does not make it a ‘social construct’.
We label rocks based on how and when they were formed. This does not mean it has any sociology behind it.
But I do respect your opinion and if you do think so, you are well within your rights to do it.
Sex is not a social construct. Gender is.
It’s true - we are learning more about sex every year and understanding it’s not completely binary. But your sex is assigned at conception.
That’s one of the reasons doctor’s don’t ask about your gender, but your sex - because treating could be different. And, as an example, IHE (medical standard) recognizes about 7 sexes.
Gender, on the other hand, defines social norms we expect from certain sex. How gender is perceived changes from culture to culture and from one period of history to another - sex doesn’t.
That being said, I do believe that gender roles should be a thing of the past. You do you, whatever you’re comfortable with.
In all seriousness, I think I’ve seen ‘Mx’ used in this context.
Usually though, you’d go with a generic greeting such as ‘Dear Sir or Madam’ or ‘To whom it may concern’.
No. They don’t and they wouldn’t.