Hard to "just ignore it" when it's in your face all the time... and yes, those who have never questioned and don't care will not notice the pronouns. For people who are in flux or who are non-conforming, it's as noticeable as someone poking you in the eye every time it happens.

My character is male, he is who he is, a separate entity from myself. But if I tried to recreate someone exactly like myself in the game I'd be in OP's position where I didn't really fit since characters would always, always, always use he/she. I'm sympathetic in general to the desire to see more than gendered pronouns for this reason. If there was a neutral option I would definitely use it! However...

Personally I don't think XIV is the place to add this stuff in. It's a game made with clear binary divisions in mind (just look at all the restricted glamours, fashions, and even hairstyles) and it wouldn't be a net benefit to the game to retrofit even something as simple as dialogue options. As many other people have said the systems behind selecting the pronoun to use are more complicated than they might seem at first glance. I'd rather see QoL upgrades to other systems like glamour, Eureka, gearing, and more.

More options are always better. But XIV is what it is at this point in terms of gender representation and I wouldn't want to change it just to make me and OP happy. That's something they can consider for a different game, if they so choose. Honestly though, even Western games haven't really come around to the concept and they're much more "ahead of the curve" so to speak than Eastern devs for this kind of representation. Just make a character you like to look at and don't try to recreate yourself to a T. It's just not possible and asking dev resources to make it work for our tiny minority is unreasonable.