In short? I bet they don't actually take it seriously enough to consider those questions carefully, but that's more of a Humanities problem than a trans problem.
The ability to experience gender dysphoria would necessitate gender being an intrinsic or at least internalized trait of the mind: otherwise there would be nothing to be dysphoric about. The
body can't feel dysphoric, only the mind can.
The theory also assumes that gender (at least the gender categories we use) are social constructs. So those traits aren't essential but contingent, and their content and meaning is only valid withing the context that gave rise to it (aka: we live in a society

). That's what “social construct” means after all.
So far, no contradiction is encountered, as we can claim one's gender is an internalized social construct trait of one's mind. Sexual characteristics are, therefore, only as relevant to gender as the society that validates them hold them to be.
This is the first trap: conflating body and this nebulous trait we haven't really even gotten any insight into because it's literally a variable and whose existence is socially dependent.
the WHO cites gender as “characteristics of women, men, girls and boys that are socially constructed” and including “norms, behaviours and roles” and “relationships with each other” and presumably themselves. You'll note that this definition carries no essence, as it basically states “gender is a partition of shit people do”.
the Canadian Institutes of Health Research defines it thus:
refers to the socially constructed roles, behaviours, expressions and identities of girls, women, boys, men, and gender diverse people. It influences how people perceive themselves and each other, how they act and interact
Again, it has no prescriptive qualities. Is “Serial Murderer” a gender? It a set of behaviors, identities and diverse peoples. It influences how people perceive themselves and are perceived by others and how they've acted and act. And how people interact with them.
Planned parenthood is more of the same, but they have a creepy line I just had to include
- Gender identity is how you feel inside and how you express your gender through clothing, behavior, and personal appearance. It’s a feeling that begins very early in life.
The best you can get is
Gender is the social relation between external signifiers and certain types of assumed fragments of identity and personality
More over, those signifiers are taken to be broadly related to sexual characteristics and fashion, and those identities, roles, behaviors and personality fragments are exceedingly generic in what they stand for.