The part people are calling a pivot is actually the part where the barrel is pinned to the stock. Many muzzle loading black powder pistols the barrels were thicker at the breech and
tapered to a quite thin wall at the muzzle. That same thickness and taper to thin is found on many muskets. I think you found a part of a muzzle loading gun, but the fact there isn't a breech plug, and no threads evident showing there could be a breech
plug, my swag is it's been sawed off a muzzle loading musket. What is the measurement of the bore? The term musket and rifle are not interchangeable, although many people will do that.
A musket has a smooth bore and is a distinctive type gun, and rifle has spiral grooves -- called rifling's -- that impart a spin to the bullet, making the weapon very accurate, and a rifle is usually distinctive in profile from a musket. Muskets are
notoriously in accurate. Just to muddy up the water, fowlers -- what we call shotguns today -- were designed for bird hunting, and are very much like a musket, but much more refined and lighter
thin barrel walls, less wood in the stock, and no feature for a bayonet. So your barrel part could have also come from a fowler.
