There actually have been several purpose-built NFA shotguns. The World's fighting Shotguns by Thomas Swearengen lists quite a few including the Ithacas, Holland Auto-Burglar Gun, The Defiance Anit-Bandit Gun P-50, Burgess Folding Police Gun.
Then there are the insurgency/counter-insurgency shotguns; Hillberg Insurgency Weapons, including the Colt Defender (eight barrels) and the Winchester Liberator series(four barrels).
Charles Manville designed a whole series of rotary chambered shotugn-style weapons, everything from 12 gauge revolvers to 24 shot 12 gauge machinegun to 18 shot 25mms. Think of the Striker 12 and Street Sweeper, only these were designed in the late 1930s.
And then there are the class III automatic shotguns; Foote Machine Shotgun, Gordon Close Support Weapons, Childers Special Operations Weapon, Atchisson Assault Shotgun (selective fire), Gilbert Assault Shotgun (USAS 12 full auto version)(BTW these are seen in the movie Tango & Cash).