I think you're very on-target here pointing out that the kind of nit-picky definitions that people try to apply are really rather ridiculous, and that language mostly doesn't work that way: a creature is a dog based on exemplars and the known diversity of things that are dogs, not based on a precise definition; likewise a chair is known by standard examples and the breadth of diversity of examples. If there are tricky cases (e.g. is a 3/4 dog, 1/4 wolf a "dog"?), people generally treat them specially and don't worry overly much about forcing the boundaries (depending on context, our one-quarter-wolf could either be called a dog or not and few people would likely bat an eye, and in any case where it really mattered, we'd use the more precise description--this is the standard way to deal with linguistic ambiguity about category boundaries: relax when you can, specify when you can't).

On the other hand, trans people and allies have also forced the issue from the other side by declaring that, for instance, "trans women are women". Not some trans women are women (e.g. those who already fit well enough within the space of examples of who are considered "women"). Not "let's call trans women 'women' when precise distinctions don't matter". Additionally, they've attacked the standard "pretty close" type of definition by denying that "woman" is a good enough approximation (in context) for "menstruator" and other such things.

So that we've ended up in this quandary of unproductive fights over strict delineations of what counts as being a "woman".

But then there is the unresolved question: if some people want to change the definition of a word, and others don't, how do you have a useful discussion about the merits? The usual way this happens is that you get linguistic subgroups (e.g. what is a "chip" vs a "crisp"), but that doesn't work here either because the argument is about what the universal usage should be.