Does this “Best” carry a meaning or it is the ultimate in self promotion akin to “Sure Shot”?
It is pretty much “self promotion akin to “Sure Shot”” as you suspected. It was used on brass shells only but meant nothing more than Express or Sure Shot. Very nice shell.
Looks like one of the WW2 and later rounds for use with military trench or riot guns. The brass cases were adopted (as the M19 if I recall correctly) late in WW2 after relearning the WW1 experience that paper shotshells do not hold up well in military service.
Earlier in WW2 the Army had used paper cased shotshells with 00 buck from many makers.
I believe that much earlier Remington had made brass cases with several names other than “BEST” but this type endured after the others were dropped, and they just kept the name because that is what they always had used.