I agree it takes a fair degree of thought to work out what's going on.
I think about it like this:
Level 1: Biology >>
signals gender and social identity.
Level 2: Psychology >>
determines gender identity and signals biological identity.
Level 3: Society >>
determines gender norms, which form the context in which gender identity is broadly interpreted.
You go from biology (the human/sex) to psychology (the person/gender identity) to the social (the group/gender norms) and you get different answers depending on how you pose the question of definition and at what level you pose it.
So, biology is relevant to gender but underdetermines it because it underdetermines (necessarily) psychology. It doesn't provide us enough evidence to assign a gender identity to a person, only to a body. This brings us back to where the denialists always end up, either explicitly or implicitly claiming the mind and society are not real and by extension that people are not real, simply because they can't accept this underdetermination and so end up reducing everything to Level 1. Animals are irrelevant because they are always stuck at Level 1, being neither people nor social in the sense we are.
So, the ground of debate is not here at the level of biology (human/animal) vs psychology (person), rather at the level of the relationship between gender identity (psychology) and gender norms (society). Minds do exist, people do exist ,and some people do identify as a gender that "contradicts" their biological sex.
Maybe the reason some can't move on from this is they misunderstand the "identity" in "gender identity" as something like preference or choice/behaviour. Language is slippery here because we don't normally need to strongly delineate between identity and preference. "I like dogs" can be phrased, "I am a dog person" {preference=identity} and that's fine. There's a degree of stickiness there. Maybe more than "I like pistachio flavor ice-cream" (as to say "I'm a pistachio-flavor-ice-cream person"{preference=identity} sounds a bit over the top). But when we say "I am a man" or "I am a woman", we're talking an identity that's core. So, it's not just that it's not a choice/behaviour, it's beyond a preference in that there's no coherent preference=identity pair to fall back on. I am a man doesn't mean I like manliness or even I like being a man, it just means that I identify as my understanding of the gender norm "man".
You can be a homosexual man, for example, who dresses and acts in a feminine way but still identifies as a man. It's not primarily about behaviour, or preference, identity is what it
means to that person to be a man or a woman and whether their understanding of themselves can be squared with society in general. Society's job then, in my view, is to be inclusive to the degree that it is rational and practical to so be. And there is nothing irrational or impractical about recognizing the phenomenon of psychological identity as it relates to gender. Just the opposite.