They were outlawed (well, more like "taxed to death", $200 was like $3000 in 1934) for cosmetic and "feel good" reasons, just as "assult weapons" are targeted today.
Shorter shotgun barrels do nothig to aid or hinder dispersal. That's determined by the type of shot, to a small degree the shot/wad, and the choke, if any.
IIRC, shot loads spread an average of 1" per yard of travel, give or take, based on the above factors. So unless you live in a mansion, any shotgun fired indoors shouldn't spread much wider than a man's outstreched hand at any interior distance inside the average home.
(Which also debunks the "don't need to aim" myth.)
So all other factors being equal, the short and long shotguns would pattern the same, minus the fraction of a meter of 1" of spread, that the differences in the barrel lengths would give. So maybe an extra .16" of spread...
My understanding of inclusion of the SBR and SBS's, was that the '34 NFA was going to make ALL HANDGUNS restricted just like Machine guns. And that the SBR and SBS categories were created to prevent people from making "handguns" out of unrestricted rifles and shotguns.
When it was realized the handgun portion of the NFA was too extreme to let it pass, it was dropped, and the SBR and SBS portions of the NFA were left in, either as oversight, or as a "crime fighting" measure.
