It's good to keep in mind that in-universe the Labyrinth's Constitution was written by someone, it didn't poof into existence out of nowhere. Someone thought out those details and made a conscious choice to phrase it that way, and that says something both about the person who wrote it and the what the times were like back then.
If the deed was passed on to a human female, or any person not born of human parents, it would not "stick". The transferal would not be effective, the previous owner would still be the de facto master, the alleged new owner would have none of the powers that come with it.
So... Someone made a conscious decision to forbid non-humans and non-men from being the Labyrinth's master. And, although that's not written in the Constitution, later masters made it even more restrictive by only passing its ownership to Catholics. You can say that's unfair and, yes, that's the point.
That said, there are confounding factors: there are plenty of loopholes for everything and the Labyrinth was created in the same age as the one where Hermaphroditus, Caeneus and Tiresias became mythological figures. Maybe the same gods who were associated with one of those figures had a hand in the labyrinth too? If so, it wouldn't be surprising if the Labyrinth has a concept of of maleness/femaleness as a state of being instead of an inborn trait. Or, even if it doesn't have that built in, maybe it can be made to think in that way.
Another thing I can add, very clearly and plainly, is that there was someone who worked in the hotel centuries ago, back when the Constitution hadn't been translated to a human readable state. This someone started presenting as a man during their stay in the hotel, and he had quite a story. He was one of the great explorers of the valley and, after many excursions, he had an epiphany.
I hope we can one day show a bit of it. He tried becoming the next master but, sadly, it became clear he would never be picked. But in the end he went on to achieve his ultimate goal, through other means.
Hopefully that sets you in the right trail, lore wise! But if that wasn't enough, I'll just say that they/them pronouns are not incompatible with being the master. That would not be an impediment...
but yeah lmao, editing the game for that would be a gargantuan task. One much harder than dealing with the lore hahaha
