This is in place of the Sudra. It is considered a Midas Chasidus but was commonly worn by all. Perhaps this is because it served the purpose of Beged Elyon as well.
People wear Tefillin without checking up the sources, too. In fact, not following the Hanhaga until you got to figure it all out on your own is not a great practice. It uproots the last remaining threads of Torah Shel Baal Peh of the original kind.