This is the flaw. It isn’t. It is required if we will write rules with respect to it. But there is another way. Forcing everything into an objective definition is the source of our divides, not a solution to anything.
Statesmen, from Cicero to Hamilton to Obama, understood this. But there is an emerging tendency to treat every system as technical, and that is destructive.
That it’s irrelevant to the definition of a woman. That’s the strength of fuzzy definitions.
When separating the kids, did anyone formally define what a girl or a boy is? Did every parent in that room need to resolve every edge case to their implied definitions ex ante? Could you guarantee conflict by forcing a formal definition on that group, even if it results in the same practical outcome the implied, unsaid definition yielded and which was peacefully accepted by the group? No, no and, of course, yes.
Now, you are right that for the purposes of this event no one defined what a girl or a boy is. The reason it is so (as is the case with many other social conventions) is because neither the participants nor the organizers challenge the classification.
There is no doubt that had it been challenged strongly and frequently enough, the implicit definitions would become explicit and formal.
By people without any civics background, yes. That we lack leaders who push back against overspecification is a cultural failing, collectively, and an individual failing among those who don’t understand the point of nuance.
It’s Cartesian, not circular. There’s a difference. We don’t need it rigorously defined for it to have meaning—that’s the point. (This does make it a word incompatible with precise endeavours like lawmaking.)