Can't help with the belt concept, but for the last couple of decades, I've used various iterations of a Game Bag to tote shells, pheasants, rabbits, ducks and squirrels in my little neck of the woods. I don't know how the technique would work for vast expanses or huge daily limits, nor, really, if you carried food stuffs for yourself while you hunted, but for a two-a-day pheasant limit, three rabbits (back when I hunted them), and three or so squirrels, a dozen or so shells, a sheath knife, a bottle of water for the dog, a steel comb, a small stash of toilet paper, and a line to use as a leash, they've worked well for me.
Now I had many hunting coats, but the weather fluctuations in the Mid-west mandated either under garments that burst the buttons, or some outer garment that covered the pockets of the hunting jacket. So, I just dressed in layers for the day and tossed the game bag over the whole ensemble.
As I've aged, I've even got separate bags for 20ga and 12ga, and for upland birds, or for waterfowl. Not because I needed them, but it was easier for me.
I offer it for your consideration. None of my bags were made for the purpose, just long strap shoulder bags from resale shops or, occasionally, from eBay.
Best wishes.
