I reckon 95% plus of "birders" do not partake in shooting, be it clays, targets, back yard plonking or hunting game (birds, mammals or fish). Where did you gather your information from to say "lots of people claiming to do both"?
I believe when people start to enjoy watching birds, butterflies, plants and nature in general most lose any desire to hunt said species and that killing instinct is absorbed or converted into other energies. I know as I was one of the few that had a healthy interest in nature as a child but also shot an air pistol, rifle and then shotgun into my early twenties. I collected bird eggs as part of that interest, and went rough shooting on farms - rabbits, woodpigeon and corvids......strictly no gamebirds was the instruction. One winter snowy day whilst out looking for rabbits on a farmer's field margins I just unloaded the gun, went home and never thought of shooting again for "sport" and gave away the gun and cartridges.
Living in a rural village in a county with many shoots, one does see a high amount of road kills of Pheasant and RLP but only this morning a Grey flew up from the verge and over the road.