I feel that 'gender' is probably a misleading term for the languages that have 'grammatical gender', it rarely has anything to do with genitalia. 'Noun class', where adjectives have to decline to agree with the class would fit better in most cases.
English essentially does not have decline adjectives, except for historical outliers like blond/e where no-one much cares if you don't bother, and uses his / hers / its / erc using a very predictable rule. So no 'grammatical gender'.
At some point it was also used as a (probably less common than it is now) synonym for sex.
It was chosen to describe the non-physical concept we now call gender exactly because the original use to describe words doesn't have anything to do with genitalia -- and the intent was that "gender" would refer only to what's between one's ears and "sex" only to what's between one's legs.
in humans the gender can be any, even when the person has specific genitalia. so saying gender is a misleading term because it rarely has to do with genitalia doesn't make much sense to me.
so basically i dont see why not just call it gender when the pronouns given to each word in such languages is gendered
English has the peculiarity of having two variants of the same word: "gender" and "genre" with slightly different meanings.
You could lean on it and go with genre. But just changing the word is unlikely to help much, the concept itself is deeply associated with genitalia in English culture, you'd still need to explain it.
The problem is that the noun class that is used to refer to you is based on your gender. As long as that is the case, grammatical gender will probably be the most apt name for the concept.