Let's re-review the physics. There is a giant difference between fins and flutes. We are discussing the properties of shallow, rounded flutes as are commonly found in sporting rifles today.
Picture a cross cut showing the diameter of a barrell. It is basically two circles (the outside barrell and the bore). In the middle we have the steel. Now divide the outside circumference into twelve equal arcs. Every other arc will become a flute - ok. Now invert the odd arcs and you'll have six flutes. Have you changed the arcs. No. You haven't even changes the surface area of the barrell, only the external shape of the barrell. That addresses the frequent misconception that a fluted barrell has more area and sheds heat faster. Remember, we are not comparing a finned barrell to a round barrell, but a fluted barrell to a round barrell and the area of our flutes is equal to the initial area of the twelve arcs (cylinder) we started with. A sporting barrel may in fact have a different arc for its flutes, but the surface area change is extremely minimal and inconsequential as a heat sink.
Now lets address the strength issue. Can any structure become stronger by removing a part of it? Picture a solid steel round rod. If you mill away the east-west sides, it is probably about as stiff in the north-south plane, but much less in the east west plane. Keep on milling the rod to any number of sides you want. As you remove material, you do not add strength. I predict that any strength test measured across the bottom of two opposing flutes, will reveal a weaker barrell in that plane . Maybe barrells should have an odd number of flutes to prevent this. Still, they will not be stronger barrells by virtue of removing any steel.
When a flute is milled into a barrell's surface, weight is removed. That is correct. A flued barrel is lighter than the blank from which it is was milled, but also heavier than a barrell equaling the diameter of the bottom of the flutes.
Flutes sell barrell and guns, mostly because they appeal to buyers. They look neat. They are new. We want them for their looks as well as any other percieved or disproven beneft.