Anyone who says round barrels weren't easier to file than barrels with flats has obviously never draw filed a barrel. Believe me, round is easier.
The problem of course is that before roughly 1860, no one had been able to figure out how to drill a long straight hole in a solid bar of iron or steel. The long drills kept wandering and breaking. Finally, someone came up with the "spoon bit" and turning the barrel rather than the drill, and the problem was solved, though most barrels still needed straightening.
Yes, building up a barrel from strips or plates wrapped around a mandrel is what we now call "Damascus" and in later years was used mainly for shotguns. But prior to that, almost all barrels were made by that general method. Some military musket barrels were made by heating a long flat of iron and then wrapping it around a mandrel lengthwise, sort of like a hot dog in a roll. The edges were then welded together. That produced a barrel that was stronger than one built up. True, the seam was a potential weak point, but with a good overlap, those barrels were quite strong.
Another method, also used with musket barrels in the Civil War era, was to use a large (3" or so) round billet that was short enough to be drilled, then insert the mandrel and pass the billet through a series of rollers of decreasing size until the barrel was formed. IIRC, Springfield Armory used that system.
Jim