It's absolutely a matter of principle, which I don't think is a surprise to you. I find that it's common to conflate refusal to oblige what amounts to policing my ability to communicate, with refusal to oblige someone's request to be addressed a certain way. One is just a matter of retaining my right to decide for myself how I communicate, but I've never had the right to say anything I want and have it not be capable of being interpreted as rude.
The thing is, you don't get to decide objectively what is wrong and right, or what I find difficult to fluently introduce into my vocabulary. A business kind of does, in so much as they can keep employees, but generally we've accepted a high bar for that, largely relying on subjectivity rather than them deciding to legislate bad words for adults.
You can't just decide that a term is objectively sexist or a personal attack, and "I really doubt" arguments aren't convincing, evidenced in part by you already implying it's anti-woman.
edit: I'd add that I'm not particularly attached to specific phrases like this, it all depends on circumstances and context, and a matter of what would make me start thinking about a change. Generally, I do think about how I'd address a group of people, because some opt not to use this or that, and it's best to find that comfortable common ground, and if you're speaking with customers I could probably more easily grasp a business wanting to pick the safest option, especially if it's likely to make it's way onto Twitter when my barista commits a faux-pas. If I worked at Reddit when they implemented their new policies for example, I'd probably start looking elsewhere. Pick your battles, and not every fight is worth your job. But if my company decides for me that this isn't an interpersonal issue, and I don't get to make these choices for myself, but that it's by definition hate speech, then maybe I'll go work on a farm or whatever you'd have me do. One that butches animals for mass human consumption; I hope that's ok ;)
This is why so many don't take you seriously.