Church etiquette on headgear has very very very long traditions which wound up being illogical.
Woman were at one time required to wear something on the head (even if it's a bobby-pinned tissue if you didn't have anything else). That harkens back to the Ancient Near East traditions of keep your women (your wife and your daughters) out of the leering eyes of other men, cf. the wedding veil and the burka.
Meanwhile, in some traditions it was absolutely blasphemous for men to wear anything in church on their heads even though the bishop was wearing two hats at the same time. Meanwhile, in the Jewish tradition, everyone covered their head.
Secular traditions are just as illogical. Not that long ago, women had to be covered from neck to wrists to ankles, even on the beach. But after their 'liberation' it was expected they wear as little as possible even to very formal events where the little black dress or the red carpet dress covered no more than a bath towel; all the while formal for men is being covered from neck to wrists to ankles...