The very point of gender is that it is socially constructed. It's a social agreement, not a biological fact. (Yes, gender is related to an underlying biological fact; no, that is not remotely relevant).
In a state of nature, no one is an accountant either, and yet we have regulations and recognition of that job.
Societies are supposed to accommodate, within reason, the people that live within them. Abstract principles are never particularly useful as guides in that situation: you figure it out on a case-by-case basis.
Accommodating people who view themselves as another gender is easy. It is not a particularly difficult challenge. Do you have to memorize the seventy or what not genders that folks have created? No. When someone asks you to accommodate their specific gender identity, is it reasonable to deny them? No. It's a simple matter of courtesy.
The science is pretty clear: gender dysmorphia is real, it has biological antecedents, and it is not an illness. People who identify as transgender should receive support to deal with the secondary symptoms that may stem from gender dysmorphia, but the underlying condition itself is not a disability or an illness. And even if it were, leprosy is undoubtedly an illness and people accommodated lepers in the past too with appropriate changes to the social contract.
There's simply no justification for choosing a priori to make some portion of the population uncomfortable. No one would want that done to them, so it is a moral grotesquerie to do it to others.