I get what you're saying, and to a degree I agree with it.

It's not that I don't care, it's that I don't want to care. I don't want to care about what genders people (in general) go by, and would rather be polite to individuals I come across by referring to them the way they ask (in the same way I refer to people who go by a shortened name or their middle name). I have not cared about high school girls sports and won't until my daughter reaches high school... or rather, I have wanted not to. It's a tiny fraction of a tiny fraction of a tiny fraction of the population, and I want their existence to not register with me until I come across them organically (e.g. a trans girl on my daughter's cross country team). I want to not know about neo-pronouns without either taking a gender studies class or meeting someone who uses them (which I have yet to in 43 years). Or if there is a trans character or one who uses neo-pronouns in a game that just is that without that being their whole character (e.g. Krem from Dragon Age Inquisition). I want those people to just exist in the world without me giving any more concern to their sense of identity or whatever as any other of the 10 billion people on the planet I don't know and don't give a shit about except they are humans and I want them to be well.

I WANT to not care, but I'm not being let to because people feel the need to pick on them, make laws against them existing, or deny them equal opportunities for spurious reasons. And every time a conservative type complains about everything trans related and having to deal with pronouns and everything else, I'm like "it's because of you! You're doing this! I wouldn't have to be concerned with them beyond knowing they exist if it weren't for you!"

I don't want to sound callous, because I do care about how LGBTQ+ people are treated. And I wish I didn't because they just existed without me having to care.

Edit: I also think people are misunderstanding your point. Like I said, I never cared about girls high school sports, so when some conservative speaking against trans girls on girls soccer teams offering some hyperbole of The Rock (or comparable AMAB player) on a girls soccer team to suggest the supposed absurdity of having a trans girl on the team, you couldn't pay me to care (about that argument). Even if it were to be the case, which it never, ever, ever is.