I love this design, it's so badass. I always picture the second version being used by power armoured soldiers in close combat as a full-auto gauss shotgun. Add the retractable bayonet from the Starcraft 2 gauss rifle just under the barrel of this, and you're ready to rock!
It's a cool concept but (No offense) it's actually impossible to belt feed shot-shells because of the way they're designed. They have this rim at the base that would lodge in the belt and prevent the round from chambering.
The soft shell isnt much of a problem, although the AA-12 does feed better with harder plastic shells. As for the lip, the AA-12 uses a mechanism that works like a semi-auto shotgun, which means the fire rate ends up pretty low, but it feed reliably. The Saiga is basicly an enlarged AK mechanism, which(typical of Soviet engineering) will feed pretty much anything reliably, with the bolt modified to accommodate the lip.
You're right. Past attempts to making a belt fed shotgun weapon were plagued by jams,not only because of the rim, but because of the plastic shells themselves. I've seen videos of amateur gunsmiths modding an AR lower to take a belt on full auto, but would have to be cleared every 3-5 rounds.
And it's a shame, because this design looks great, and would have made for a cool alternative.