why exactly should I care which sex the person checking me in at the hotel thinks I look like?
This is implausibly stupid. Of course it would affect a trans person differently to a cis person! For hugely obvious reasons.
Yes, and to get to a mentally healthy state, they need to work on that - it's not healthy to be so affected by a random person, performing a task completely unrelated to sex.
Pffft. There is nothing inherent about gendered pronouns corresponding with a sex. "Grammatical gender" is part of the social construct rather than the biological fact
Yes, exactly - but words - which are all socially constructed - mean things, that's the social convention we have, that when we say chair, we mean that thing with legs and a back which we sit on. When we say 'the feminine grammatical gender' in reference to an animal or a human we are referring to someone/something that is female.
If the feminine grammatical gender corresponds instead with the feminine social gender then a) people have been misgendering me all my life, and b) we're going to need to be able to define the feminine social gender, and c) there's going to be a lot more judgement around as people try to guess whether to use he or she based on what shoes someone's wearing!
If words don't mean things, then we have no point of reference to work on.