“The Shulchan Aruch agrees that a “chassid” should wear both pairs.”
Doesn’t the word chassid used in the Sh”A refer to people who are mekayeim the mitzvos with a measure of chassidus, meaning EXTRA-devotion and are machmirim (as opposed to the way in which we use the term “chassidus” today to refer to people who follow a rebbie such as Bobov, Lubavitch, Satmar, etc.? If so, then it has nothing to do with the latter at all.