It's gonna blow your mind when you find out that many trans women are describing themselves as female now.
And this is the crux of the matter really.
It doesn't matter what words we use. The ultimate goal is to replace biological sex with socially constructed gender in all circumstances. In order to do that, all the words we have which mean biological sex must be redefined to mean socially constructed gender.
Trans women have already claimed the word "woman" and are now working on "female", which means that your suggestion of abandoning the words "man" and "woman" and simply using "male" and "female" is already out of date.
Let's take this one step further. Say we abandon the words "woman" and "female" and come up with an entirely new word to mean "adult humans of the biological sex which, in the absence of any disorders of sexual development or other infertility, is potentially capable of gestating, birthing and breastfeeding human infants between the menarche and the menopause".
What could this new word be? Let's say we go with "quinglet", which I came up with using a random word generator I found online.
About five minutes after we start using the word "quinglet" to refer to the group of people currently understood to be adult humans of the female biological sex, this is what will happen. Some adult humans of the male biological sex will start claiming to identify as that, and some adult humans of the female biological sex will start complaining that they don't identify as that, and asking us not to use it in a way that includes them.
This is how we have ended up with any male person being able to use the words "woman" and "female" to describe himself, whilst female people are redefined as "menstruators" and "people with cervixes" and "pregnant people" and "birthing bodies" to accommodate a small group of female people who, having asked us not to use the word "women" to include them, now want us to come up with euphemisms for female people so that they can be included.
I'm sorry, but these people need to be told "no". We need clear vocabulary to identify and discuss biological sex in humans. That means we need a word for all humans of the childbearing sex, which includes trans men and female bodied non binary people but not trans women. And we need a word for all humans of the sperm producing sex. I don't care whether you don't want to be included within the group that you are in, or you want to be included within the group that you are not in. I don't care. The criteria for inclusion are objective, not subjective. You do not get to dictate that the rest of society should not have words for concepts that it needs to be able to talk about just because those concepts offend you.
We don't need new vocabulary. We just need to reclaim the vocabulary that already exists and restore it to its original meaning.
Men who believe they identify as whatever it is they believe they are identifying with can come up with a new word for that and use it to refer to themselves. Women who happily refer to themselves as "cis women" and believe that they share some kind of identity with these men can also use that new word to refer to themselves if they wish.
I believe "quinglet" is available.