There was a meshugana troll some time ago who has left now, but one of his crazy nonsense was a Halachic piece where he suggested that it was a chumra and that to wear a brim wider than a tefach brim-down was assur on Shabbos.
But he was crazy, and trying his best to be a troll.
A lot of the reason is to do with the fact that lots of Bochurim use their old Shabbos hat for weekdays, and is often not in such a good shape, and doesn’t look so great with the brim down. With the brim up looks better.