Why do you think it's incoherent ...
Because the words are commonly used in many different and quite contradictory ways. I see that Elaedith quoted a Singal article in a recent comment that underlines those ambiguities, inconsistencies, and "mission-critical" contradictions:
Gender-affirming care is medically necessary ... care ... to help a person transition from their assigned gender — the one the person was designated at birth — to their affirmed gender — the gender by which one wants to be known.
Rank flaming insanity. Being charitable. To a fault. Jesus, Himself, would be hard-pressed to do better.
NO ONE changes their sex which is basically what they're arguing or what their dogma boils down into. NO ONE gets to exchange their ovaries for testicles or vice versa. NO ONE is assigned a gender at birth. Does the midwife put on a sorting hat and says, "hmm, clearly destined to be an introvert, or maybe an
INFP or possibly an ESTJ ..."?
Because that is basically what gender boils down into -- sexually dimorphic personality traits. The conflation of those traits with the underlying traits that define the sex categories is the problem -- as I've indicated here several times.
Clearly differentiating between the two sets of traits -- between those, on the one hand, that are intrinsic or essential to the whole process of reproduction, and those, on the other hand, that are not, but which correlate with the first set -- is what seems an essential step in resolving the whole problem. Which has a great many quite serious consequences. Philosopher Will Durant on Voltaire:
Durant: 'If you wish to converse with me,' said Voltaire, 'define your terms.' How many a debate would have been deflated into a paragraph if the disputants had dared to define their terms! This is the alpha and omega of logic, the heart and soul of it, that every important term in serious discourse shall be subjected to strictest scrutiny and definition. It is difficult, and ruthlessly tests the mind; but once done it is half of any task."
https://quotefancy.com/quote/300152...me-said-Voltaire-define-your-terms-How-many-a
... and why then do you keep putting the word incoherent in scare quotes?
Being charitable; being kind; trying -- is spite of extreme provocation ... -- to be "civil" ...

I don't think we need the word gender at all to be applied to humans. In most of our history, it hasn't been used that way and I don't see anything gained by it.
Oxford Dictionaries -- arguably a if not
the gold standard -- has chapter and verse showing that that is not at all the case:
https://www.oed.com/dictionary/gender_n
Moot exactly how "gender" and "sex" have been used in all of those cases -- and I really don't have the time or the linguistics background to get very far into the question. But there's some reason to argue that, as I've argued above, the former were related to "accidental properties" and the latter to "essential properties" of the sex categories defined on the basis of actual, not potential, abilities to reproduce.
https://plato.stanford.edu/entries/essential-accidental/
On the contrary, I see a lot of attempts at sleight of hand and attempts to gain power through language based upon the word "gender" as applied to humans. What would we lose if we did away with it?
A great deal, a massive amount. For one thing, the ability to differentiate between those accidental and essential properties.
Those sexually dimorphic traits -- particularly in personality traits, behaviours, roles, and predilections -- seem ubiquitous and of more than passing import. See this article in
4th Wave Now, one of the co-authors being Colin Wright:
https://4thwavenow.com/2019/08/19/n...r-thoughts-on-the-concept-of-gender-identity/
Of particular note is a graph of a two dimensional joint probability distribution that emphasizes the wide differences in the average values of many traits between men and women. As Helen Dale has emphasized, evolution didn't stop at our necks. Notwithstanding the bogus assertions of various "feminists" to the contrary.
I see nothing added by using the word "gender" and lots of negatives and obfuscation.
No doubt many are trying to make political hay out of that obfuscation. Notably in using "male" and "female" as genders -- why I've repeatedly insisted on using "masculine" and "feminine" as genders, and on reserving "male" and "female" to denote the sexes as transitory reproductive abilities:
Anne transitioned from male to female in her 40s.
https://en.wikipedia.org/wiki/Anne_Lawrence
In 2012, Hubbard transitioned to female. She began hormone therapy that year. Hubbard competed in international weightlifting for the first time in 2017
https://en.wikipedia.org/wiki/Laurel_Hubbard
https://humanuseofhumanbeings.substack.com/p/wikipedias-lysenkoism
And why I've expended some effort in trying to put gender -- as some set of those sexually dimorphic personality traits -- on something of a sound scientific footing, at least as a way of separating wheat and chaff:
https://humanuseofhumanbeings.substack.com/p/a-multi-dimensional-gender-spectrum
Sexual dimorphism really is a ubiquitous phenomenon, and it extends far beyond simple physical traits or "phenotypes". Seems of more than passing value and import to understand the roots of it -- which I think will be impossible if we abandon both the term "gender" and the profound differences between it and "sex":
https://en.wikipedia.org/wiki/Sexual_dimorphism
https://en.wikipedia.org/wiki/Gender