Maybe this analogy will help: Let’s say you meet someone, and you ask them “Do you have a cat?”. Note that you’ve used the singular here, though it’s acting number-neutral in this context. If they respond “I have two”, then it will immediately become ungrammatical for you to continue to use the number-neutral singular and ask “Does your cat like fish?”

This helped a lot. It is true that I do not feel this way about "they", but it does put things into perspective. Thank you.

That doesn’t mean that we can’t, or shouldn’t, try to accommodate non-binary people of course, as is unfortunately often argued, but it does mean that, contrary to what I commonly see people say on the internet, doing so for these speakers does require a constant, concerted effort to consciously override their mental grammars.

This is true also when someone you've known for a while transitions, or changes their name (like in the case of marriage, or just a regular name-change). Most people are okay with you tripping up, it's expected even. It's just when it's done in bad faith that it becomes an issue.