A few thoughts...
A society where people could change their sex down to their DNA would be radically different from our own. I can see it playing out two ways...
Either the society, like our own, would have one sex which has a real socialisation, career progression, and financial advantage, so everyone would want to choose to be that sex. This would create similar social issues to those we currently have in societies where there is sex-selection abortion, but on a much grander scale. To address this, there would have to be limits on who could change sex, and how. And maybe markers to say 'this one's not a real man' to keep uppity women out of male spaces.
Or, where the burdens of childrearing can fall equally on anyone, then sex-based power differentials could melt away. I can only see that happening if there's an element of choicelessness on the sex change.
If you want to see a thought-experiment played out on the latter, then I would highly recommend Ursula Le Guin's 'The Left Hand of Darkness'. It's one of my very favourite books, and one of its themes is just that.
In either event, it would be a radically different society from our own. So what I, as an individual socialised in a society where I can't change my sex, no matter how much I would want to, would think of individuals who can change their sex at will really isn't the point. The point is how that society would manage when sex is a fluid concept.
I agree it would be fascinating. It would allow us to unpick what is socialisation, what is male-reinforced power structures, and what is the inevitable consequences of a binary sexual biology. And whether there really is a thing called gender identity that exists and persists independent of biological sex. But what I think of it, as a creature who cannot escape my female biology, is irrelevant.