Prior to current military vests (Level IIIA with a level III or IV full frontal plate), vests served not so much to stop actual slugs shot at soldiers, but the other types of potential damage from shrapnel, ricochets, and non-projectile shrapnel such as wood spliters from trees or rock splinters from brick, rock, mortar, etc. In that case, a vest might be helpful for protecting against secondary types of wounding and you would not need more than soft armor (highest rated is IIIA).
For bird hunting, armor would likely be a nice addition. Unless up really close, the armor should stop shotgun BBs and pellets that had lost enough velocity down to around no more than 1200 fps, or pistol velocity range.
Regardless of ballistic protection, for winter hunting, a ballistic vest offers something else as well, WARMTH! They can be quite toasty.
Would a ballistic vest be helpful to hunters? Probably not as much as you might think unless they work the full Level IV stuff that is pretty bulky and can be uncomfortable. Even then, it is limited in the protection to being just the body. A local kid here was shot by a 30.06 by his uncle while he and his father were approaching the uncle's tree blind. The kid was lucky that all that resulted was a mangled nasty looking arm. It functions, but isn't pretty. The idiot uncle, realizing they had run out of time and still had yet to land a deer shot at movement in the brush. The uncle never actually had a visual on an animal, otherwise he would have realized the movement was by the orange vest wearing nephew and brother.