In my main language, Dutch, we don't even have separate words for the two concepts. We just have 'man' (man) and 'vrouw' (woman), which refer to sex in humans, but also that of other species. For example, we'd say 'het is een vrouwtje' (it's a female) when talking about a dog. Talking about 'female women' and 'male women' simply doesn't work in most languages outside of English because there's no distinction possible.

On top of that, we already have the words for gender that aren't 'man' and 'woman'. It's 'masculine' and 'feminine'. The terms 'man' and 'woman' have always referred to sex as well, never gender.