If "women" no longer means "biologically female people" and "men" no longer means "biologically male people" then both words just mean "some unspecified people". There is no difference between the two groups which can be defined by reference to objective criteria.
So, as you say, why have different provisions for men and women at all?
As a PP said, it then becomes necessary to re-examine the purpose of these things.
Take sports, for example. Why do we have two sporting categories? Answer: because humans come in two different types, one of which has a body which is, on average, much bigger and more powerful than that of the other type, to the point where if those two types of humans did not have their own categories, the type with the smaller, less powerful bodies would not be able to compete in sport at all because even the best of them are not competitive against the type with the bigger, more powerful bodies. Clearly these two body types relate to biology, not identity. This means that we need sporting categories classified according to these two biological types, and we also need clear vocabulary to name and describe those two biological types, so we know who is eligible to compete in which category.
If you essentially expunge the words "men" and "women" from the English language on the grounds that they now both mean "some people" and are therefore no longer useful to us in terms of differentiating between different types or humans, and then you identify all situations in which we formerly had separate provision for "men" and "women" and analyse the purpose of that separate provision, why it is needed and what it is based on, you will end up right back where we started. Which is here:
In all situations where western society has separate provision for men and women, the reason for that separate provision is based on biology, not identity. In order for this to work we need to have terms for people of each biological sex, and for those terms not to be used by members of the opposite biological sex to describe their subjectively experienced gender identity.
(I say western society because there are some cultures in which men and women are separated for reasons I do not think are legitimate, e.g. in Afghanistan.)