My understanding is that these days, we are not allowed to opine on whether others are wrong to feel offended. In modern parlance, I'd say that you don't understand why they're offended. It seems you're trying very hard not to, considering that this term is not the preferred term of reference for practically anyone who is cisgender.
If the best you can do is tell me that I have to accede to the ineffable and indefinite reasons for your neighbor taking offense, then I suppose the discussion is over. (Of course I do accede in practice - I’m not going to refer to her as ‘they’ just to be a dick.)
If you really believe that people are entitled to be arbitrarily offended by pronouns, then you can have no objection to trans or non-binary people being arbitrarily fussy about their use. I mean, I’m “woke” by HN standards and am all in favor of respecting people’s pronoun preferences. But even I don’t believe that the subject is entirely in the realm of subjective feelings of offense and beyond rational discourse.
For this reason I think that people who are offended by being referred to by 'they' are wrong to be offended. I think they're wrong in a way that, say, a trans woman is not wrong to be offended if someone insists on using 'he'. In the former case, the person may feel that they are being misgendered – but only on the basis of a dodgy linguistic analysis. Of course, we should accommodate people's pronoun preferences, regardless of whether we agree with their underlying logic in any given case.
The story about your neighbor doesn't really make sense to me. Surely it occurred to her that the other forum poster might not have cared much what her gender was, or have paid much attention to her photo, or simply didn't care to take a guess even if she appeared clearly female. To interpret this as intentional misgendering seems a bit nuts. If she was simply worried about whether she appeared clearly feminine in the photo, then that's an understandable anxiety, but one that has little to do with singular they. (If singular they were off limits then I guess the poster might have used 'he' instead, which hardly seems better.)