Society has long relied on the role differences between genders. How useful are they? 100% of societies have them, with only a scant few new cultures (like ours) trying to see what happens as we break them down.
Why are they useful? Consider what happens when you ask a 5 year old what they want to be when they grow up. The answer is usually something like "an astronaut" or "a firefighter." That answer will change over time, so its not really worth giving a 5 year old astronaut training (other than that, perhaps, its fun as all heck for the 5 year old!)
Now, consider a role which benefits from lifelong training. A culture which begins training at 7 is fundamentally 7 years behind a culture that trains at birth in such a role. Now consider if there are diametrically opposed roles, where it is terribly hard to be good at both. There would be an advantage to artificially dividing the culture in half and giving each half training from birth.
Societies have shown that, with the presence of biological sex, that division is incredibly beneficial in a remarkable number of situations. Only recently have we explored whether giving individuals more choice and less sex-driven roles is beneficial for society or not (and the experiment is still running!).
So, in your case with a fluid biological sex, the answer is less obvious. It could be that this results in a society with fewer roles being forced on us, and more choice for individuals. This is the utopia side of things. It's also possible that it is tremendously difficult to keep up without such a split of roles, and society crumbles under it (the distopia side of things).
More likely, however, society would not crumble. Society dislikes doing things like that. More likely, if it needed a division of roles from birth to succeed, it would develop a new method of division. Perhaps birth gender might be tatooed on you. Or perhaps society would divide on hair color. Or perhaps a powerful caste system world form that divides the world into two castes that always intermarry.
This is what I find incredibly exciting about the recent shifts in women's roles. We, as a society, actually have absolutely no clue what the result will be. All we can do is go down a path, and see what the answer is. Hopefully we can go at a pace which allows us to retain balance as we go!