(I'm guessing that you're going to find this upsetting and insulting?)

No, but I find it so unrealistc that it is boringly easy to answer. And is it really so unclear what I find upsetting and insulting in this debate, and why? (That's a question. If I get a "yes" for an answer, I can clarify)

You are back at reducing the headscarf (which lurks behind your ominous hat) to a symbol of obeisance to authoritarianism, and the act wearing them as unvoluntary and enforced, and all the the girls who do as victims without agency. We have been here before. First of all, Islam (and religion in general) is not authoritarian. Some practices are. In those cases where compulsion plays a role, the ban on headscarves doesn't solve the problem. In the better case of compulsion you haven't altered the situation, in the worse case you harden positions of parents and girls. Some girls consent to wearing headscarves in order to achieve more freedom in other fields, and a ban on this strategy increases their problems. In many other cases there is no compulsion, and you are banning girls from wearing a piece of clothing that is important or even essential for practising their religion (if you want to harp on your non-religious group: for a social network they attach importance to), or that is important for them for other reasons. One motivation for wearing a headscarf which you consistently ignore or ridicule is setting a counterpoint to the compulsion to objectifying clothing, by the way.

So much for your thought experiment. You can't claim it was realistical, can you? If you have a phobia against hats, do something about it. If you can't see that there are many reasons to wear a headscarf or a hat, you are blind to reality. And if you want to do something against authoritarianism, fight authoritarianism.